Linear Algebra

2021

• Paper 1, Section I, $1 \mathrm{E}$

  Linear Algebra | Part IB, 2021

  Let $V$ be a vector space over $\mathbb{R}, \operatorname{dim} V=n$, and let $\langle,$,$rangle be a non-degenerate anti-$ symmetric bilinear form on $V$.

  Let $v \in V, v \neq 0$. Show that $v^{\perp}$ is of dimension $n-1$ and $v \in v^{\perp}$. Show that if $W \subseteq v^{\perp}$ is a subspace with $W \oplus \mathbb{R} v=v^{\perp}$, then the restriction of $\langle,$,$rangle to W$ is nondegenerate.

  Conclude that the dimension of $V$ is even.

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• Paper 1, Section II, E

  Linear Algebra | Part IB, 2021

  Let $d \geqslant 1$, and let $J_{d}=\left(\begin{array}{ccccc}0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ & & \cdots & \cdots & \\ 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \ldots & 0 & 0\end{array}\right) \in \operatorname{Mat}_{d}(\mathbb{C})$.

  (a) (i) Compute $J_{d}^{n}$, for all $n \geqslant 0$.

  (ii) Hence, or otherwise, compute $\left(\lambda I+J_{d}\right)^{n}$, for all $n \geqslant 0$.

  (b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $\varphi \in \operatorname{End}(V)$. Suppose $\varphi^{n}=0$ for some $n>1$.

  (i) Determine the possible eigenvalues of $\varphi$.

  (ii) What are the possible Jordan blocks of $\varphi$ ?

  (iii) Show that if $\varphi^{2}=0$, there exists a decomposition

  $$V=U \oplus W_{1} \oplus W_{2}$$

  where $\varphi(U)=\varphi\left(W_{1}\right)=0, \varphi\left(W_{2}\right)=W_{1}$, and $\operatorname{dim} W_{2}=\operatorname{dim} W_{1}$.

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• Paper 2, Section II, E

  Linear Algebra | Part IB, 2021

  (a) Compute the characteristic polynomial and minimal polynomial of

  $$A=\left(\begin{array}{ccc} -2 & -6 & -9 \\ 3 & 7 & 9 \\ -1 & -2 & -2 \end{array}\right)$$

  Write down the Jordan normal form for $A$.

  (b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}, f: V \rightarrow V$ be a linear map, and for $\alpha \in \mathbb{C}, n \geqslant 1$, write

  $$W_{\alpha, n}:=\left\{v \in V \mid(f-\alpha I)^{n} v=0\right\}$$

  (i) Given $v \in W_{\alpha, n}, v \neq 0$, construct a non-zero eigenvector for $f$ in terms of $v$.

  (ii) Show that if $w_{1}, \ldots, w_{d}$ are non-zero eigenvectors for $f$ with eigenvalues $\alpha_{1}, \ldots, \alpha_{d}$, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $w_{1}, \ldots, w_{d}$ are linearly independent.

  (iii) Show that if $v_{1} \in W_{\alpha_{1}, n}, \ldots, v_{d} \in W_{\alpha_{d}, n}$ are all non-zero, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $v_{1}, \ldots, v_{d}$ are linearly independent.

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• Paper 3, Section II, 9E

  Linear Algebra | Part IB, 2021

  (a) (i) State the rank-nullity theorem.

  Let $U$ and $W$ be vector spaces. Write down the definition of their direct sum $U \oplus W$ and the inclusions $i: U \rightarrow U \oplus W, j: W \rightarrow U \oplus W$.

  Now let $U$ and $W$ be subspaces of a vector space $V$. Define $l: U \cap W \rightarrow U \oplus W$ by $l(x)=i x-j x .$

  Describe the quotient space $(U \oplus W) / \operatorname{Im}(l)$ as a subspace of $V$.

  (ii) Let $V=\mathbb{R}^{5}$, and let $U$ be the subspace of $V$ spanned by the vectors

  $$\left(\begin{array}{c} 1 \\ 2 \\ -1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{c} -2 \\ 2 \\ 2 \\ 1 \\ -2 \end{array}\right)$$

  and $W$ the subspace of $V$ spanned by the vectors

  $$\left(\begin{array}{c} 3 \\ 2 \\ -3 \\ 1 \\ 3 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{c} 1 \\ -4 \\ -1 \\ -2 \\ 1 \end{array}\right)$$

  Determine the dimension of $U \cap W$.

  (b) Let $A, B$ be complex $n$ by $n$ matrices with $\operatorname{rank}(B)=k$.

  Show that $\operatorname{det}(A+t B)$ is a polynomial in $t$ of degree at most $k$.

  Show that if $k=n$ the polynomial is of degree precisely $n$.

  Give an example where $k \geqslant 1$ but this polynomial is zero.

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• Paper 4, Section I, $1 \mathbf{E}$

  Linear Algebra | Part IB, 2021

  Let $\operatorname{Mat}_{n}(\mathbb{C})$ be the vector space of $n$ by $n$ complex matrices.

  Given $A \in \operatorname{Mat}_{n}(\mathbb{C})$, define the linear $\operatorname{map}_{A}: \operatorname{Mat}_{n}(\mathbb{C}) \rightarrow \operatorname{Mat}_{n}(\mathbb{C})$,

  $$X \mapsto A X-X A$$

  (i) Compute a basis of eigenvectors, and their associated eigenvalues, when $A$ is the diagonal matrix

  $$A=\left(\begin{array}{llll} 1 & & & \\ & 2 & & \\ & & \ddots & \\ & & & n \end{array}\right)$$

  What is the rank of $\varphi_{A}$ ?

  (ii) Now let $A=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$. Write down the matrix of the linear transformation $\varphi_{A}$ with respect to the standard basis of $\operatorname{Mat}_{2}(\mathbb{C})$.

  What is its Jordan normal form?

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• Paper 4, Section II, E

  Linear Algebra | Part IB, 2021

  (a) Let $V$ be a complex vector space of dimension $n$.

  What is a Hermitian form on $V$ ?

  Given a Hermitian form, define the matrix $A$ of the form with respect to the basis $v_{1}, \ldots, v_{n}$ of $V$, and describe in terms of $A$ the value of the Hermitian form on two elements of $V$.

  Now let $w_{1}, \ldots, w_{n}$ be another basis of $V$. Suppose $w_{i}=\sum_{j} p_{i j} v_{j}$, and let $P=\left(p_{i j}\right)$. Write down the matrix of the form with respect to this new basis in terms of $A$ and $P$.

  Let $N=V^{\perp}$. Describe the dimension of $N$ in terms of the matrix $A$.

  (b) Write down the matrix of the real quadratic form

  $$x^{2}+y^{2}+2 z^{2}+2 x y+2 x z-2 y z .$$

  Using the Gram-Schmidt algorithm, find a basis which diagonalises the form. What are its rank and signature?

  (c) Let $V$ be a real vector space, and $\langle,$,$rangle a symmetric bilinear form on it. Let A$ be the matrix of this form in some basis.

  Prove that the signature of $\langle,$,$rangle is the number of positive eigenvalues of A$ minus the number of negative eigenvalues.

  Explain, using an example, why the eigenvalues themselves depend on the choice of a basis.

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2020

• Paper 1, Section I, F

  Linear Algebra | Part IB, 2020

  Define what it means for two $n \times n$ matrices $A$ and $B$ to be similar. Define the Jordan normal form of a matrix.

  Determine whether the matrices

  $$A=\left(\begin{array}{ccc} 4 & 6 & -15 \\ 1 & 3 & -5 \\ 1 & 2 & -4 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & -3 & 3 \\ -2 & -6 & 13 \\ -1 & -4 & 8 \end{array}\right)$$

  are similar, carefully stating any theorem you use.

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• Paper 1, Section II, F

  Linear Algebra | Part IB, 2020

  Let $\mathcal{M}_{n}$ denote the vector space of $n \times n$ matrices over a field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. What is the $\operatorname{rank} r(A)$ of a matrix $A \in \mathcal{M}_{n}$ ?

  Show, stating accurately any preliminary results that you require, that $r(A)=n$ if and only if $A$ is non-singular, i.e. $\operatorname{det} A \neq 0$.

  Does $\mathcal{M}_{n}$ have a basis consisting of non-singular matrices? Justify your answer.

  Suppose that an $n \times n$ matrix $A$ is non-singular and every entry of $A$ is either 0 or 1. Let $c_{n}$ be the largest possible number of 1 's in such an $A$. Show that $c_{n} \leqslant n^{2}-n+1$. Is this bound attained? Justify your answer.

  [Standard properties of the adjugate matrix can be assumed, if accurately stated.]

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• Paper 2, Section II, F

  Linear Algebra | Part IB, 2020

  Let $V$ be a finite-dimensional vector space over a field. Show that an endomorphism $\alpha$ of $V$ is idempotent, i.e. $\alpha^{2}=\alpha$, if and only if $\alpha$ is a projection onto its image.

  Determine whether the following statements are true or false, giving a proof or counterexample as appropriate:

  (i) If $\alpha^{3}=\alpha^{2}$, then $\alpha$ is idempotent.

  (ii) The condition $\alpha(1-\alpha)^{2}=0$ is equivalent to $\alpha$ being idempotent.

  (iii) If $\alpha$ and $\beta$ are idempotent and such that $\alpha+\beta$ is also idempotent, then $\alpha \beta=0$.

  (iv) If $\alpha$ and $\beta$ are idempotent and $\alpha \beta=0$, then $\alpha+\beta$ is also idempotent.

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2019

• Paper 1, Section I, F

  Linear Algebra | Part IB, 2019

  Define a basis of a vector space $V$.

  If $V$ has a finite basis $\mathcal{B}$, show using only the definition that any other basis $\mathcal{B}^{\prime}$ has the same cardinality as $\mathcal{B}$.

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• Paper 1, Section II, F

  Linear Algebra | Part IB, 2019

  What is the adjugate adj $(A)$ of an $n \times n$ matrix $A$ ? How is it related to $\operatorname{det}(A) ?$

  (a) Define matrices $B_{0}, B_{1}, \ldots, B_{n-1}$ by

  $$\operatorname{adj}(t I-A)=\sum_{i=0}^{n-1} B_{i} t^{n-1-i}$$

  and scalars $c_{0}, c_{1}, \ldots, c_{n}$ by

  $$\operatorname{det}(t I-A)=\sum_{j=0}^{n} c_{j} t^{n-j}$$

  Find a recursion for the matrices $B_{i}$ in terms of $A$ and the $c_{j}$ 's.

  (b) By considering the partial derivatives of the multivariable polynomial

  $$p\left(t_{1}, t_{2}, \ldots, t_{n}\right)=\operatorname{det}\left(\left(\begin{array}{cccc} t_{1} & 0 & \cdots & 0 \\ 0 & t_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & t_{n} \end{array}\right)-A\right)$$

  show that

  $$\frac{d}{d t}(\operatorname{det}(t I-A))=\operatorname{Tr}(\operatorname{adj}(t I-A))$$

  (c) Hence show that the $c_{j}$ 's may be expressed in terms of $\operatorname{Tr}(A), \operatorname{Tr}\left(A^{2}\right), \ldots, \operatorname{Tr}\left(A^{n}\right)$.

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• Paper 2, Section I, F

  Linear Algebra | Part IB, 2019

  If $U$ and $W$ are finite-dimensional subspaces of a vector space $V$, prove that

  $$\operatorname{dim}(U+W)=\operatorname{dim}(U)+\operatorname{dim}(W)-\operatorname{dim}(U \cap W)$$

  Let

  $$\begin{aligned} U &=\left\{\mathbf{x} \in \mathbb{R}^{4} \mid x_{1}=7 x_{3}+8 x_{4}, x_{2}+5 x_{3}+6 x_{4}=0\right\} \\ W &=\left\{\mathbf{x} \in \mathbb{R}^{4} \mid x_{1}+2 x_{2}+3 x_{3}=0, x_{4}=0\right\} . \end{aligned}$$

  Show that $U+W$ is 3 -dimensional and find a linear map $\ell: \mathbb{R}^{4} \rightarrow \mathbb{R}$ such that

  $$U+W=\left\{\mathbf{x} \in \mathbb{R}^{4} \mid \ell(\mathbf{x})=0\right\}$$

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• Paper 2, Section II, F

  Linear Algebra | Part IB, 2019

  Let $A$ and $B$ be $n \times n$ matrices over $\mathbb{C}$.

  (a) Assuming that $A$ is invertible, show that $A B$ and $B A$ have the same characteristic polynomial.

  (b) By considering the matrices $A-s I$, show that $A B$ and $B A$ have the same characteristic polynomial even when $A$ is singular.

  (c) Give an example to show that the minimal polynomials $m_{A B}(t)$ and $m_{B A}(t)$ of $A B$ and $B A$ may be different.

  (d) Show that $m_{A B}(t)$ and $m_{B A}(t)$ differ at most by a factor of $t$. Stating carefully any results which you use, deduce that if $A B$ is diagonalisable then so is $(B A)^{2}$.

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• Paper 3, Section II, F

  Linear Algebra | Part IB, 2019

  If $q$ is a quadratic form on a finite-dimensional real vector space $V$, what is the associated symmetric bilinear form $\varphi(\cdot, \cdot)$ ? Prove that there is a basis for $V$ with respect to which the matrix for $\varphi$ is diagonal. What is the signature of $q$ ?

  If $R \leqslant V$ is a subspace such that $\varphi(r, v)=0$ for all $r \in R$ and all $v \in V$, show that $q^{\prime}(v+R)=q(v)$ defines a quadratic form on the quotient vector space $V / R$. Show that the signature of $q^{\prime}$ is the same as that of $q$.

  If $e, f \in V$ are vectors such that $\varphi(e, e)=0$ and $\varphi(e, f)=1$, show that there is a direct sum decomposition $V=\operatorname{span}(e, f) \oplus U$ such that the signature of $\left.q\right|_{U}$ is the same as that of $q$.

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• Paper 4, Section I, F

  Linear Algebra | Part IB, 2019

  What is an eigenvalue of a matrix $A$ ? What is the eigenspace corresponding to an eigenvalue $\lambda$ of $A$ ?

  Consider the matrix

  $$A=\left(\begin{array}{cccc} a a & a b & a c & a d \\ b a & b b & b c & b d \\ c a & c b & c c & c d \\ d a & d b & d c & d d \end{array}\right)$$

  for $(a, b, c, d) \in \mathbb{R}^{4}$ a non-zero vector. Show that $A$ has rank 1 . Find the eigenvalues of $A$ and describe the corresponding eigenspaces. Is $A$ diagonalisable?

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• Paper 4, Section II, F

  Linear Algebra | Part IB, 2019

  If $U$ is a finite-dimensional real vector space with inner product $\langle\cdot, \cdot\rangle$, prove that the linear map $\phi: U \rightarrow U^{*}$ given by $\phi(u)\left(u^{\prime}\right)=\left\langle u, u^{\prime}\right\rangle$ is an isomorphism. [You do not need to show that it is linear.]

  If $V$ and $W$ are inner product spaces and $\alpha: V \rightarrow W$ is a linear map, what is meant by the adjoint $\alpha^{*}$ of $\alpha$ ? If $\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}$ is an orthonormal basis for $V,\left\{f_{1}, f_{2}, \ldots, f_{m}\right\}$ is an orthonormal basis for $W$, and $A$ is the matrix representing $\alpha$ in these bases, derive a formula for the matrix representing $\alpha^{*}$ in these bases.

  Prove that $\operatorname{Im}(\alpha)=\operatorname{Ker}\left(\alpha^{*}\right)^{\perp}$.

  If $w_{0} \notin \operatorname{Im}(\alpha)$ then the linear equation $\alpha(v)=w_{0}$ has no solution, but we may instead search for a $v_{0} \in V$ minimising $\left\|\alpha(v)-w_{0}\right\|^{2}$, known as a least-squares solution. Show that $v_{0}$ is such a least-squares solution if and only if it satisfies $\alpha^{*} \alpha\left(v_{0}\right)=\alpha^{*}\left(w_{0}\right)$. Hence find a least-squares solution to the linear equation

  $$\left(\begin{array}{ll} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right)$$

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2018

• Paper 1, Section I, E

  Linear Algebra | Part IB, 2018

  State the Rank-Nullity Theorem.

  If $\alpha: V \rightarrow W$ and $\beta: W \rightarrow X$ are linear maps and $W$ is finite dimensional, show that

  $$\operatorname{dim} \operatorname{Im}(\alpha)=\operatorname{dim} \operatorname{Im}(\beta \alpha)+\operatorname{dim}(\operatorname{Im}(\alpha) \cap \operatorname{Ker}(\beta))$$

  If $\gamma: U \rightarrow V$ is another linear map, show that

  $$\operatorname{dim} \operatorname{Im}(\beta \alpha)+\operatorname{dim} \operatorname{Im}(\alpha \gamma) \leqslant \operatorname{dim} \operatorname{Im}(\alpha)+\operatorname{dim} \operatorname{Im}(\beta \alpha \gamma)$$

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• Paper 1, Section II, E

  Linear Algebra | Part IB, 2018

  Define a Jordan block $J_{m}(\lambda)$. What does it mean for a complex $n \times n$ matrix to be in Jordan normal form?

  If $A$ is a matrix in Jordan normal form for an endomorphism $\alpha: V \rightarrow V$, prove that

  $$\operatorname{dim} \operatorname{Ker}\left((\alpha-\lambda I)^{r}\right)-\operatorname{dim} \operatorname{Ker}\left((\alpha-\lambda I)^{r-1}\right)$$

  is the number of Jordan blocks $J_{m}(\lambda)$ of $A$ with $m \geqslant r$.

  Find a matrix in Jordan normal form for $J_{m}(\lambda)^{2}$. [Consider all possible values of $\lambda$.]

  Find a matrix in Jordan normal form for the complex matrix

  $$\left[\begin{array}{cccc} 0 & 0 & 0 & a_{1} \\ 0 & 0 & a_{2} & 0 \\ 0 & a_{3} & 0 & 0 \\ a_{4} & 0 & 0 & 0 \end{array}\right]$$

  assuming it is invertible.

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• Paper 2, Section I, E

  Linear Algebra | Part IB, 2018

  Let $V$ be a real vector space. Define the dual vector space $V^{*}$ of $V$. If $U$ is a subspace of $V$, define the annihilator $U^{0}$ of $U$. If $x_{1}, x_{2}, \ldots, x_{n}$ is a basis for $V$, define its dual $x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*}$ and prove that it is a basis for $V^{*}$.

  If $V$ has basis $x_{1}, x_{2}, x_{3}, x_{4}$ and $U$ is the subspace spanned by

  $$x_{1}+2 x_{2}+3 x_{3}+4 x_{4} \quad \text { and } \quad 5 x_{1}+6 x_{2}+7 x_{3}+8 x_{4},$$

  give a basis for $U^{0}$ in terms of the dual basis $x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, x_{4}^{*}$.

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• Paper 2, Section II, E

  Linear Algebra | Part IB, 2018

  If $X$ is an $n \times m$ matrix over a field, show that there are invertible matrices $P$ and $Q$ such that

  $$Q^{-1} X P=\left[\begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array}\right]$$

  for some $0 \leqslant r \leqslant \min (m, n)$, where $I_{r}$ is the identity matrix of dimension $r$.

  For a square matrix of the form $A=\left[\begin{array}{cc}B & D \\ 0 & C\end{array}\right]$ with $B$ and $C$ square matrices, prove that $\operatorname{det}(A)=\operatorname{det}(B) \operatorname{det}(C)$.

  If $A \in M_{n \times n}(\mathbb{C})$ and $B \in M_{m \times m}(\mathbb{C})$ have no common eigenvalue, show that the linear map

  $$\begin{aligned} L: M_{n \times m}(\mathbb{C}) & \longrightarrow M_{n \times m}(\mathbb{C}) \\ X & \longmapsto A X-X B \end{aligned}$$

  is injective.

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• Paper 3, Section II, E

  Linear Algebra | Part IB, 2018

  State and prove the Cayley-Hamilton Theorem.

  Let $A$ be an $n \times n$ complex matrix. Using division of polynomials, show that if $p(x)$ is a polynomial then there is another polynomial $r(x)$ of degree at most $(n-1)$ such that $p(\lambda)=r(\lambda)$ for each eigenvalue $\lambda$ of $A$ and such that $p(A)=r(A)$.

  Hence compute the $(1,1)$ entry of the matrix $A^{1000}$ when

  $$A=\left[\begin{array}{ccc} 2 & -1 & 0 \\ 1 & -1 & 1 \\ -1 & -1 & 1 \end{array}\right]$$

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• Paper 4, Section I, E

  Linear Algebra | Part IB, 2018

  Define a quadratic form on a finite dimensional real vector space. What does it mean for a quadratic form to be positive definite?

  Find a basis with respect to which the quadratic form

  $$x^{2}+2 x y+2 y^{2}+2 y z+3 z^{2}$$

  is diagonal. Is this quadratic form positive definite?

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• Paper 4, Section II, E

  Linear Algebra | Part IB, 2018

  Let $V$ be a finite dimensional inner-product space over $\mathbb{C}$. What does it mean to say that an endomorphism of $V$ is self-adjoint? Prove that a self-adjoint endomorphism has real eigenvalues and may be diagonalised.

  An endomorphism $\alpha: V \rightarrow V$ is called positive definite if it is self-adjoint and satisfies $\langle\alpha(x), x\rangle>0$ for all non-zero $x \in V$; it is called negative definite if $-\alpha$ is positive definite. Characterise the property of being positive definite in terms of eigenvalues, and show that the sum of two positive definite endomorphisms is positive definite.

  Show that a self-adjoint endomorphism $\alpha: V \rightarrow V$ has all eigenvalues in the interval $[a, b]$ if and only if $\alpha-\lambda I$ is positive definite for all $\lambda<a$ and negative definite for all $\lambda>b$.

  Let $\alpha, \beta: V \rightarrow V$ be self-adjoint endomorphisms whose eigenvalues lie in the intervals $[a, b]$ and $[c, d]$ respectively. Show that all of the eigenvalues of $\alpha+\beta$ lie in the interval $[a+c, b+d]$.

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2017

• Paper 1, Section I, F

  Linear Algebra | Part IB, 2017

  State and prove the Steinitz Exchange Lemma.

  Deduce that, for a subset $S$ of $\mathbb{R}^{n}$, any two of the following imply the third:

  (i) $S$ is linearly independent

  (ii) $S$ is spanning

  (iii) $S$ has exactly $n$ elements

  Let $e_{1}, e_{2}$ be a basis of $\mathbb{R}^{2}$. For which values of $\lambda$ do $\lambda e_{1}+e_{2}, e_{1}+\lambda e_{2}$ form a basis of $\mathbb{R}^{2} ?$

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• Paper 1, Section II, F

  Linear Algebra | Part IB, 2017

  Let $U$ and $V$ be finite-dimensional real vector spaces, and let $\alpha: U \rightarrow V$ be a surjective linear map. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.

  (i) There is a linear map $\beta: V \rightarrow U$ such that $\beta \alpha$ is the identity map on $U$.

  (ii) There is a linear map $\beta: V \rightarrow U$ such that $\alpha \beta$ is the identity map on $V$.

  (iii) There is a subspace $W$ of $U$ such that the restriction of $\alpha$ to $W$ is an isomorphism from $W$ to $V$.

  (iv) If $X$ and $Y$ are subspaces of $U$ with $U=X \oplus Y$ then $V=\alpha(X) \oplus \alpha(Y)$.

  (v) If $X$ and $Y$ are subspaces of $U$ with $V=\alpha(X) \oplus \alpha(Y)$ then $U=X \oplus Y$.

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• Paper 2, Section I, F

  Linear Algebra | Part IB, 2017

  State and prove the Rank-Nullity theorem.

  Let $\alpha$ be a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ of rank 2 . Give an example to show that $\mathbb{R}^{3}$ may be the direct sum of the kernel of $\alpha$ and the image of $\alpha$, and also an example where this is not the case.

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• Paper 2, Section II, F

  Linear Algebra | Part IB, 2017

  Let $\alpha: U \rightarrow V$ and $\beta: V \rightarrow W$ be linear maps between finite-dimensional real vector spaces.

  Show that the rank $r(\beta \alpha)$ satisfies $r(\beta \alpha) \leqslant \min (r(\beta), r(\alpha))$. Show also that $r(\beta \alpha) \geqslant r(\alpha)+r(\beta)-\operatorname{dim} V$. For each of these two inequalities, give examples to show that we may or may not have equality.

  Now let $V$ have dimension $2 n$ and let $\alpha: V \rightarrow V$ be a linear map of rank $2 n-2$ such that $\alpha^{n}=0$. Find the rank of $\alpha^{k}$ for each $1 \leqslant k \leqslant n-1$.

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• Paper 3, Section II, F

  Linear Algebra | Part IB, 2017

  Let $f$ be a quadratic form on a finite-dimensional real vector space $V$. Prove that there exists a diagonal basis for $f$, meaning a basis with respect to which the matrix of $f$ is diagonal.

  Define the rank $r$ and signature $s$ of $f$ in terms of this matrix. Prove that $r$ and $s$ are independent of the choice of diagonal basis.

  In terms of $r, s$, and the dimension $n$ of $V$, what is the greatest dimension of a subspace on which $f$ is zero?

  Now let $f$ be the quadratic form on $\mathbb{R}^{3}$ given by $f(x, y, z)=x^{2}-y^{2}$. For which points $v$ in $\mathbb{R}^{3}$ is it the case that there is some diagonal basis for $f$ containing $v$ ?

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• Paper 4, Section I, F

  Linear Algebra | Part IB, 2017

  Briefly explain the Gram-Schmidt orthogonalisation process in a real finite-dimensional inner product space $V$.

  For a subspace $U$ of $V$, define $U^{\perp}$, and show that $V=U \oplus U^{\perp}$.

  For which positive integers $n$ does

  $$(f, g)=f(1) g(1)+f(2) g(2)+f(3) g(3)$$

  define an inner product on the space of all real polynomials of degree at most $n$ ?

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• Paper 4, Section II, F

  Linear Algebra | Part IB, 2017

  What is the dual $X^{*}$ of a finite-dimensional real vector space $X$ ? If $X$ has a basis $e_{1}, \ldots, e_{n}$, define the dual basis, and prove that it is indeed a basis of $X^{*}$.

  [No results on the dimension of duals may be assumed without proof.]

  Write down (without making a choice of basis) an isomorphism from $X$ to $X^{* *}$. Prove that your map is indeed an isomorphism.

  Does every basis of $X^{*}$ arise as the dual basis of some basis of $X ?$ Justify your answer.

  A subspace $W$ of $X^{*}$ is called separating if for every non-zero $x \in X$ there is a $T \in W$ with $T(x) \neq 0$. Show that the only separating subspace of $X^{*}$ is $X^{*}$ itself.

  Now let $X$ be the (infinite-dimensional) space of all real polynomials. Explain briefly how we may identify $X^{*}$ with the space of all real sequences. Give an example of a proper subspace of $X^{*}$ that is separating.

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2016

• Paper 1, Section $I$, $1 F$

  Linear Algebra | Part IB, 2016

  (a) Consider the linear transformation $\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by the matrix

  $$\left(\begin{array}{rrr} 5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4 \end{array}\right)$$

  Find a basis of $\mathbb{R}^{3}$ in which $\alpha$ is represented by a diagonal matrix.

  (b) Give a list of $6 \times 6$ matrices such that any linear transformation $\beta: \mathbb{R}^{6} \rightarrow \mathbb{R}^{6}$ with characteristic polynomial

  $$(x-2)^{4}(x+7)^{2}$$

  and minimal polynomial

  $$(x-2)^{2}(x+7)$$

  is similar to one of the matrices on your list. No two distinct matrices on your list should be similar. [No proof is required.]

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• Paper 1, Section II, F

  Linear Algebra | Part IB, 2016

  Let $M_{n, n}$ denote the vector space over $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Let $\operatorname{Tr}: M_{n, n} \rightarrow F$ denote the trace functional, i.e., if $A=\left(a_{i j}\right)_{1 \leqslant i, j \leqslant n} \in M_{n, n}$, then

  $$\operatorname{Tr}(A)=\sum_{i=1}^{n} a_{i i}$$

  (a) Show that Tr is a linear functional.

  (b) Show that $\operatorname{Tr}(A B)=\operatorname{Tr}(B A)$ for $A, B \in M_{n, n}$.

  (c) Show that $\operatorname{Tr}$ is unique in the following sense: If $f: M_{n, n} \rightarrow F$ is a linear functional such that $f(A B)=f(B A)$ for each $A, B \in M_{n, n}$, then $f$ is a scalar multiple of the trace functional. If, in addition, $f(I)=n$, then $f=$ Tr.

  (d) Let $W \subseteq M_{n, n}$ be the subspace spanned by matrices $C$ of the form $C=A B-B A$ for $A, B \in M_{n, n}$. Show that $W$ is the kernel of Tr.

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• Paper 2, Section I, F

  Linear Algebra | Part IB, 2016

  Find a linear change of coordinates such that the quadratic form

  $$2 x^{2}+8 x y-6 x z+y^{2}-4 y z+2 z^{2}$$

  takes the form

  $$\alpha x^{2}+\beta y^{2}+\gamma z^{2}$$

  for real numbers $\alpha, \beta$ and $\gamma$.

  comment

• Paper 2, Section II, F

  Linear Algebra | Part IB, 2016

  Let $M_{n, n}$ denote the vector space over a field $F=\mathbb{R}$ or $\mathbb{C}$ of $n \times n$ matrices with entries in $F$. Given $B \in M_{n, n}$, consider the two linear transformations $R_{B}, L_{B}: M_{n, n} \rightarrow$ $M_{n, n}$ defined by

  $$L_{B}(A)=B A, \quad R_{B}(A)=A B$$

  (a) Show that $\operatorname{det} L_{B}=(\operatorname{det} B)^{n}$.

  [For parts (b) and (c), you may assume the analogous result $\operatorname{det} R_{B}=(\operatorname{det} B)^{n}$ without proof.]

  (b) Now let $F=\mathbb{C}$. For $B \in M_{n, n}$, write $B^{*}$ for the conjugate transpose of $B$, i.e., $B^{*}:=\bar{B}^{T}$. For $B \in M_{n, n}$, define the linear transformation $M_{B}: M_{n, n} \rightarrow M_{n, n}$ by

  $$M_{B}(A)=B A B^{*}$$

  Show that $\operatorname{det} M_{B}=|\operatorname{det} B|^{2 n}$.

  (c) Again let $F=\mathbb{C}$. Let $W \subseteq M_{n, n}$ be the set of Hermitian matrices. [Note that $W$ is not a vector space over $\mathbb{C}$ but only over $\mathbb{R} .]$ For $B \in M_{n, n}$ and $A \in W$, define $T_{B}(A)=B A B^{*}$. Show that $T_{B}$ is an $\mathbb{R}$-linear operator on $W$, and show that as such,

  $$\operatorname{det} T_{B}=|\operatorname{det} B|^{2 n}$$

  comment

• Paper 3, Section II, F

  Linear Algebra | Part IB, 2016

  Let $\alpha: V \rightarrow V$ be a linear transformation defined on a finite dimensional inner product space $V$ over $\mathbb{C}$. Recall that $\alpha$ is normal if $\alpha$ and its adjoint $\alpha^{*}$ commute. Show that $\alpha$ being normal is equivalent to each of the following statements:

  (i) $\alpha=\alpha_{1}+i \alpha_{2}$ where $\alpha_{1}, \alpha_{2}$ are self-adjoint operators and $\alpha_{1} \alpha_{2}=\alpha_{2} \alpha_{1}$;

  (ii) there is an orthonormal basis for $V$ consisting of eigenvectors of $\alpha$;

  (iii) there is a polynomial $g$ with complex coefficients such that $\alpha^{*}=g(\alpha)$.

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• Paper 4, Section I, F

  Linear Algebra | Part IB, 2016

  For which real numbers $x$ do the vectors

  $$(x, 1,1,1), \quad(1, x, 1,1), \quad(1,1, x, 1), \quad(1,1,1, x),$$

  not form a basis of $\mathbb{R}^{4}$ ? For each such value of $x$, what is the dimension of the subspace of $\mathbb{R}^{4}$ that they span? For each such value of $x$, provide a basis for the spanned subspace, and extend this basis to a basis of $\mathbb{R}^{4}$.

  comment

• Paper 4, Section II, F

  Linear Algebra | Part IB, 2016

  (a) Let $\alpha: V \rightarrow W$ be a linear transformation between finite dimensional vector spaces over a field $F=\mathbb{R}$ or $\mathbb{C}$.

  Define the dual map of $\alpha$. Let $\delta$ be the dual map of $\alpha$. Given a subspace $U \subseteq V$, define the annihilator $U^{\circ}$ of $U$. Show that $(\operatorname{ker} \alpha)^{\circ}$ and the image of $\delta$ coincide. Conclude that the dimension of the image of $\alpha$ is equal to the dimension of the image of $\delta$. Show that $\operatorname{dim} \operatorname{ker}(\alpha)-\operatorname{dim} \operatorname{ker}(\delta)=\operatorname{dim} V-\operatorname{dim} W$.

  (b) Now suppose in addition that $V, W$ are inner product spaces. Define the adjoint $\alpha^{*}$ of $\alpha$. Let $\beta: U \rightarrow V, \gamma: V \rightarrow W$ be linear transformations between finite dimensional inner product spaces. Suppose that the image of $\beta$ is equal to the kernel of $\gamma$. Then show that $\beta \beta^{*}+\gamma^{*} \gamma$ is an isomorphism.

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2015

• Paper 1, Section I, E

  Linear Algebra | Part IB, 2015

  Let $U$ and $V$ be finite dimensional vector spaces and $\alpha: U \rightarrow V$ a linear map. Suppose $W$ is a subspace of $U$. Prove that

  $$r(\alpha) \geqslant r\left(\left.\alpha\right|_{W}\right) \geqslant r(\alpha)-\operatorname{dim}(U)+\operatorname{dim}(W)$$

  where $r(\alpha)$ denotes the rank of $\alpha$ and $\left.\alpha\right|_{W}$ denotes the restriction of $\alpha$ to $W$. Give examples showing that each inequality can be both a strict inequality and an equality.

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• Paper 1, Section II, E

  Linear Algebra | Part IB, 2015

  Determine the characteristic polynomial of the matrix

  $$M=\left(\begin{array}{cccc} x & 1 & 1 & 0 \\ 1-x & 0 & -1 & 0 \\ 2 & 2 x & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  For which values of $x \in \mathbb{C}$ is $M$ invertible? When $M$ is not invertible determine (i) the Jordan normal form $J$ of $M$, (ii) the minimal polynomial of $M$.

  Find a basis of $\mathbb{C}^{4}$ such that $J$ is the matrix representing the endomorphism $M: \mathbb{C}^{4} \rightarrow \mathbb{C}^{4}$ in this basis. Give a change of basis matrix $P$ such that $P^{-1} M P=J$.

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• Paper 2, Section I, $1 \mathrm{E}$

  Linear Algebra | Part IB, 2015

  Let $q$ denote a quadratic form on a real vector space $V$. Define the rank and signature of $q$.

  Find the rank and signature of the following quadratic forms. (a) $q(x, y, z)=x^{2}+y^{2}+z^{2}-2 x z-2 y z$. (b) $q(x, y, z)=x y-x z$.

  (c) $q(x, y, z)=x y-2 z^{2}$.

  comment

• Paper 2, Section II, E

  Linear Algebra | Part IB, 2015

  (i) Suppose $A$ is a matrix that does not have $-1$ as an eigenvalue. Show that $A+I$ is non-singular. Further, show that $A$ commutes with $(A+I)^{-1}$.

  (ii) A matrix $A$ is called skew-symmetric if $A^{T}=-A$. Show that a real skewsymmetric matrix does not have $-1$ as an eigenvalue.

  (iii) Suppose $A$ is a real skew-symmetric matrix. Show that $U=(I-A)(I+A)^{-1}$ is orthogonal with determinant 1 .

  (iv) Verify that every orthogonal matrix $U$ with determinant 1 which does not have $-1$ as an eigenvalue can be expressed as $(I-A)(I+A)^{-1}$ where $A$ is a real skew-symmetric matrix.

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• Paper 3, Section II, E

  Linear Algebra | Part IB, 2015

  Let $A_{1}, A_{2}, \ldots, A_{k}$ be $n \times n$ matrices over a field $\mathbb{F}$. We say $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable if there exists an invertible matrix $P$ such that $P^{-1} A_{i} P$ is diagonal for all $1 \leqslant i \leqslant k$. We say the matrices are commuting if $A_{i} A_{j}=A_{j} A_{i}$ for all $i, j$.

  (i) Suppose $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable. Prove that they are commuting.

  (ii) Define an eigenspace of a matrix. Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting $n \times n$ matrices over a field $\mathbb{F}$. Let $E$ denote an eigenspace of $B_{1}$. Prove that $B_{i}(E) \leqslant E$ for all $i$.

  (iii) Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting diagonalisable matrices. Prove that they are simultaneously diagonalisable.

  (iv) Are the $2 \times 2$ diagonalisable matrices over $\mathbb{C}$ simultaneously diagonalisable? Explain your answer.

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• Paper 4, Section I, E

  Linear Algebra | Part IB, 2015

  Define the dual space $V^{*}$ of a vector space $V$. Given a basis $\left\{x_{1}, \ldots, x_{n}\right\}$ of $V$ define its dual and show it is a basis of $V^{*}$.

  Let $V$ be a 3-dimensional vector space over $\mathbb{R}$ and let $\left\{\zeta_{1}, \zeta_{2}, \zeta_{3}\right\}$ be the basis of $V^{*}$ dual to the basis $\left\{x_{1}, x_{2}, x_{3}\right\}$ for $V$. Determine, in terms of the $\zeta_{i}$, the bases dual to each of the following: (a) $\left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}\right\}$, (b) $\left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{1}\right\}$.

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• Paper 4, Section II, E

  Linear Algebra | Part IB, 2015

  Suppose $U$ and $W$ are subspaces of a vector space $V$. Explain what is meant by $U \cap W$ and $U+W$ and show that both of these are subspaces of $V$.

  Show that if $U$ and $W$ are subspaces of a finite dimensional space $V$ then

  $$\operatorname{dim} U+\operatorname{dim} W=\operatorname{dim}(U \cap W)+\operatorname{dim}(U+W)$$

  Determine the dimension of the subspace $W$ of $\mathbb{R}^{5}$ spanned by the vectors

  $$\left(\begin{array}{c} 1 \\ 3 \\ 3 \\ -1 \\ 1 \end{array}\right),\left(\begin{array}{l} 4 \\ 1 \\ 3 \\ 2 \\ 1 \end{array}\right),\left(\begin{array}{l} 3 \\ 2 \\ 1 \\ 2 \\ 3 \end{array}\right),\left(\begin{array}{c} 2 \\ 2 \\ 5 \\ -1 \\ -1 \end{array}\right)$$

  Write down a $5 \times 5$ matrix which defines a linear map $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ with $(1,1,1,1,1)^{T}$ in the kernel and with image $W$.

  What is the dimension of the space spanned by all linear maps $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$

  (i) with $(1,1,1,1,1)^{T}$ in the kernel and with image contained in $W$,

  (ii) with $(1,1,1,1,1)^{T}$ in the kernel or with image contained in $W$ ?

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2014

• Paper 1, Section I, G

  Linear Algebra | Part IB, 2014

  State and prove the Steinitz Exchange Lemma. Use it to prove that, in a finitedimensional vector space: any two bases have the same size, and every linearly independent set extends to a basis.

  Let $e_{1}, \ldots, e_{n}$ be the standard basis for $\mathbb{R}^{n}$. Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{1}$ a basis for $\mathbb{R}^{3} ?$ Is $e_{1}+e_{2}, e_{2}+e_{3}, e_{3}+e_{4}, e_{4}+e_{1}$ a basis for $\mathbb{R}^{4} ?$ Justify your answers.

  comment

• Paper 1, Section II, G

  Linear Algebra | Part IB, 2014

  Let $V$ be an $n$-dimensional real vector space, and let $T$ be an endomorphism of $V$. We say that $T$ acts on a subspace $W$ if $T(W) \subset W$.

  (i) For any $x \in V$, show that $T$ acts on the linear span of $\left\{x, T(x), T^{2}(x), \ldots, T^{n-1}(x)\right\}$.

  (ii) If $\left\{x, T(x), T^{2}(x), \ldots, T^{n-1}(x)\right\}$ spans $V$, show directly (i.e. without using the CayleyHamilton Theorem) that $T$ satisfies its own characteristic equation.

  (iii) Suppose that $T$ acts on a subspace $W$ with $W \neq\{0\}$ and $W \neq V$. Let $e_{1}, \ldots, e_{k}$ be a basis for $W$, and extend to a basis $e_{1}, \ldots, e_{n}$ for $V$. Describe the matrix of $T$ with respect to this basis.

  (iv) Using (i), (ii) and (iii) and induction, give a proof of the Cayley-Hamilton Theorem.

  [Simple properties of determinants may be assumed without proof.]

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• Paper 2, Section I, G

  Linear Algebra | Part IB, 2014

  State and prove the Rank-Nullity Theorem.

  Let $\alpha$ be a linear map from $\mathbb{R}^{5}$ to $\mathbb{R}^{3}$. What are the possible dimensions of the kernel of $\alpha$ ? Justify your answer.

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• Paper 2, Section II, G

  Linear Algebra | Part IB, 2014

  Define the determinant of an $n \times n$ complex matrix $A$. Explain, with justification, how the determinant of $A$ changes when we perform row and column operations on $A$.

  Let $A, B, C$ be complex $n \times n$ matrices. Prove the following statements. (i) $\operatorname{det}\left(\begin{array}{cc}A & C \\ 0 & B\end{array}\right)=\operatorname{det} A \operatorname{det} B$. (ii) $\operatorname{det}\left(\begin{array}{cc}A & -B \\ B & A\end{array}\right)=\operatorname{det}(A+i B) \operatorname{det}(A-i B)$.

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• Paper 3, Section II, G

  Linear Algebra | Part IB, 2014

  Let $q$ be a nonsingular quadratic form on a finite-dimensional real vector space $V$. Prove that we may write $V=P \bigoplus N$, where the restriction of $q$ to $P$ is positive definite, the restriction of $q$ to $N$ is negative definite, and $q(x+y)=q(x)+q(y)$ for all $x \in P$ and $y \in N$. [No result on diagonalisability may be assumed.]

  Show that the dimensions of $P$ and $N$ are independent of the choice of $P$ and $N$. Give an example to show that $P$ and $N$ are not themselves uniquely defined.

  Find such a decomposition $V=P \bigoplus N$ when $V=\mathbb{R}^{3}$ and $q$ is the quadratic form $q((x, y, z))=x^{2}+2 y^{2}-2 x y-2 x z$

  comment

• Paper 4, Section I, G

  Linear Algebra | Part IB, 2014

  Let $V$ denote the vector space of all real polynomials of degree at most 2 . Show that

  $$(f, g)=\int_{-1}^{1} f(x) g(x) d x$$

  defines an inner product on $V$.

  Find an orthonormal basis for $V$.

  comment

• Paper 4, Section II, G

  Linear Algebra | Part IB, 2014

  Let $V$ be a real vector space. What is the dual $V^{*}$ of $V ?$ If $e_{1}, \ldots, e_{n}$ is a basis for $V$, define the dual basis $e_{1}^{*}, \ldots, e_{n}^{*}$ for $V^{*}$, and show that it is indeed a basis for $V^{*}$.

  [No result about dimensions of dual spaces may be assumed.]

  For a subspace $U$ of $V$, what is the annihilator of $U$ ? If $V$ is $n$-dimensional, how does the dimension of the annihilator of $U$ relate to the dimension of $U$ ?

  Let $\alpha: V \rightarrow W$ be a linear map between finite-dimensional real vector spaces. What is the dual map $\alpha^{*}$ ? Explain why the rank of $\alpha^{*}$ is equal to the rank of $\alpha$. Prove that the kernel of $\alpha^{*}$ is the annihilator of the image of $\alpha$, and also that the image of $\alpha^{*}$ is the annihilator of the kernel of $\alpha$.

  [Results about the matrices representing a map and its dual may be used without proof, provided they are stated clearly.]

  Now let $V$ be the vector space of all real polynomials, and define elements $L_{0}, L_{1}, \ldots$ of $V^{*}$ by setting $L_{i}(p)$ to be the coefficient of $X^{i}$ in $p$ (for each $p \in V$ ). Do the $L_{i}$ form a basis for $V^{*}$ ?

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2013

• Paper 1, Section I, E

  Linear Algebra | Part IB, 2013

  What is the adjugate of an $n \times n$ matrix $A$ ? How is it related to $A^{-1}$ ? Suppose all the entries of $A$ are integers. Show that all the entries of $A^{-1}$ are integers if and only if $\operatorname{det} A=\pm 1$.

  comment

• Paper 1, Section II, E

  Linear Algebra | Part IB, 2013

  If $V_{1}$ and $V_{2}$ are vector spaces, what is meant by $V_{1} \oplus V_{2}$ ? If $V_{1}$ and $V_{2}$ are subspaces of a vector space $V$, what is meant by $V_{1}+V_{2}$ ?

  Stating clearly any theorems you use, show that if $V_{1}$ and $V_{2}$ are subspaces of a finite dimensional vector space $V$, then

  $$\operatorname{dim} V_{1}+\operatorname{dim} V_{2}=\operatorname{dim}\left(V_{1} \cap V_{2}\right)+\operatorname{dim}\left(V_{1}+V_{2}\right)$$

  Let $V_{1}, V_{2} \subset \mathbb{R}^{4}$ be subspaces with bases

  $$\begin{gathered} V_{1}=\langle(3,2,4,-1),(1,2,1,-2),(-2,3,3,2)\rangle \\ V_{2}=\langle(1,4,2,4),(-1,1,-1,-1),(3,1,2,0)\rangle . \end{gathered}$$

  Find a basis $\left\langle\mathbf{v}_{1}, \mathbf{v}_{2}\right\rangle$ for $V_{1} \cap V_{2}$ such that the first component of $\mathbf{v}_{1}$ and the second component of $\mathbf{v}_{2}$ are both 0 .

  comment

• Paper 2, Section I, E

  Linear Algebra | Part IB, 2013

  If $A$ is an $n \times n$ invertible Hermitian matrix, let

  $$U_{A}=\left\{U \in M_{n \times n}(\mathbb{C}) \mid \bar{U}^{T} A U=A\right\}$$

  Show that $U_{A}$ with the operation of matrix multiplication is a group, and that det $U$ has norm 1 for any $U \in U_{A}$. What is the relation between $U_{A}$ and the complex Hermitian form defined by $A$ ?

  If $A=I_{n}$ is the $n \times n$ identity matrix, show that any element of $U_{A}$ is diagonalizable.

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• Paper 2, Section II, E

  Linear Algebra | Part IB, 2013

  Define what it means for a set of vectors in a vector space $V$ to be linearly dependent. Prove from the definition that any set of $n+1$ vectors in $\mathbb{R}^{n}$ is linearly dependent.

  Using this or otherwise, prove that if $V$ has a finite basis consisting of $n$ elements, then any basis of $V$ has exactly $n$ elements.

  Let $V$ be the vector space of bounded continuous functions on $\mathbb{R}$. Show that $V$ is infinite dimensional.

  comment

• Paper 3, Section II, E

  Linear Algebra | Part IB, 2013

  Let $V$ and $W$ be finite dimensional real vector spaces and let $T: V \rightarrow W$ be a linear map. Define the dual space $V^{*}$ and the dual map $T^{*}$. Show that there is an isomorphism $\iota: V \rightarrow\left(V^{*}\right)^{*}$ which is canonical, in the sense that $\iota \circ S=\left(S^{*}\right)^{*} \circ \iota$ for any automorphism $S$ of $V$.

  Now let $W$ be an inner product space. Use the inner product to show that there is an injective map from im $T$ to $\operatorname{im} T^{*}$. Deduce that the row rank of a matrix is equal to its column rank.

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• Paper 4, Section I, E

  Linear Algebra | Part IB, 2013

  What is a quadratic form on a finite dimensional real vector space $V$ ? What does it mean for two quadratic forms to be isomorphic (i.e. congruent)? State Sylvester's law of inertia and explain the definition of the quantities which appear in it. Find the signature of the quadratic form on $\mathbb{R}^{3}$ given by $q(\mathbf{v})=\mathbf{v}^{T} A \mathbf{v}$, where

  $$A=\left(\begin{array}{ccc} -2 & 1 & 6 \\ 1 & -1 & -3 \\ 6 & -3 & 1 \end{array}\right)$$

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• Paper 4, Section II, E

  Linear Algebra | Part IB, 2013

  What does it mean for an $n \times n$ matrix to be in Jordan form? Show that if $A \in M_{n \times n}(\mathbb{C})$ is in Jordan form, there is a sequence $\left(A_{m}\right)$ of diagonalizable $n \times n$ matrices which converges to $A$, in the sense that the $(i j)$ th component of $A_{m}$ converges to the $(i j)$ th component of $A$ for all $i$ and $j$. [Hint: A matrix with distinct eigenvalues is diagonalizable.] Deduce that the same statement holds for all $A \in M_{n \times n}(\mathbb{C})$.

  Let $V=M_{2 \times 2}(\mathbb{C})$. Given $A \in V$, define a linear map $T_{A}: V \rightarrow V$ by $T_{A}(B)=A B+B A$. Express the characteristic polynomial of $T_{A}$ in terms of the trace and determinant of $A$. [Hint: First consider the case where $A$ is diagonalizable.]

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2012

• Paper 1, Section I, F

  Linear Algebra | Part IB, 2012

  Define the notions of basis and dimension of a vector space. Prove that two finitedimensional real vector spaces with the same dimension are isomorphic.

  In each case below, determine whether the set $S$ is a basis of the real vector space $V:$

  (i) $V=\mathbb{C}$ is the complex numbers; $S=\{1, i\}$.

  (ii) $V=\mathbb{R}[x]$ is the vector space of all polynomials in $x$ with real coefficients; $S=\{1,(x-1),(x-1)(x-2),(x-1)(x-2)(x-3), \ldots\} .$

  (iii) $V=\{f:[0,1] \rightarrow \mathbb{R}\} ; S=\left\{\chi_{p} \mid p \in[0,1]\right\}$, where

  $$\chi_{p}(x)= \begin{cases}1 & x=p \\ 0 & x \neq p\end{cases}$$

  comment

• Paper 1, Section II, F

  Linear Algebra | Part IB, 2012

  Define what it means for two $n \times n$ matrices to be similar to each other. Show that if two $n \times n$ matrices are similar, then the linear transformations they define have isomorphic kernels and images.

  If $A$ and $B$ are $n \times n$ real matrices, we define $[A, B]=A B-B A$. Let

  $$\begin{aligned} K_{A} &=\left\{X \in M_{n \times n}(\mathbb{R}) \mid[A, X]=0\right\} \\ L_{A} &=\left\{[A, X] \mid X \in M_{n \times n}(\mathbb{R})\right\} \end{aligned}$$

  Show that $K_{A}$ and $L_{A}$ are linear subspaces of $M_{n \times n}(\mathbb{R})$. If $A$ and $B$ are similar, show that $K_{A} \cong K_{B}$ and $L_{A} \cong L_{B}$.

  Suppose that $A$ is diagonalizable and has characteristic polynomial

  $$\left(x-\lambda_{1}\right)^{m_{1}}\left(x-\lambda_{2}\right)^{m_{2}}$$

  where $\lambda_{1} \neq \lambda_{2}$. What are $\operatorname{dim} K_{A}$ and $\operatorname{dim} L_{A} ?$

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• Paper 2, Section I, $1 F$

  Linear Algebra | Part IB, 2012

  Define the determinant $\operatorname{det} A$ of an $n \times n$ real matrix $A$. Suppose that $X$ is a matrix with block form

  $$X=\left(\begin{array}{cc} A & B \\ 0 & C \end{array}\right) \text {, }$$

  where $A, B$ and $C$ are matrices of dimensions $n \times n, n \times m$ and $m \times m$ respectively. Show that $\operatorname{det} X=(\operatorname{det} A)(\operatorname{det} C)$.

  comment

• Paper 2, Section II, F

  Linear Algebra | Part IB, 2012

  (i) Define the transpose of a matrix. If $V$ and $W$ are finite-dimensional real vector spaces, define the dual of a linear map $T: V \rightarrow W$. How are these two notions related?

  Now suppose $V$ and $W$ are finite-dimensional inner product spaces. Use the inner product on $V$ to define a linear map $V \rightarrow V^{*}$ and show that it is an isomorphism. Define the adjoint of a linear map $T: V \rightarrow W$. How are the adjoint of $T$ and its dual related? If $A$ is a matrix representing $T$, under what conditions is the adjoint of $T$ represented by the transpose of $A$ ?

  (ii) Let $V=C[0,1]$ be the vector space of continuous real-valued functions on $[0,1]$, equipped with the inner product

  $$\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t$$

  Let $T: V \rightarrow V$ be the linear map

  $$T f(t)=\int_{0}^{t} f(s) d s$$

  What is the adjoint of $T ?$

  comment

• Paper 3, Section II, F

  Linear Algebra | Part IB, 2012

  What is meant by the Jordan normal form of an $n \times n$ complex matrix?

  Find the Jordan normal forms of the following matrices:

  $$\left(\begin{array}{llll} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \quad\left(\begin{array}{cccc} -1 & -1 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \end{array}\right), \quad\left(\begin{array}{cccc} 3 & 0 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 9 & 6 & 3 & 0 \\ 15 & 12 & 9 & 3 \end{array}\right)$$

  Suppose $A$ is an invertible $n \times n$ complex matrix. Explain how to derive the characteristic and minimal polynomials of $A^{n}$ from the characteristic and minimal polynomials of $A$. Justify your answer. [Hint: write each polynomial as a product of linear factors.]

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• Paper 4, Section I, F

  Linear Algebra | Part IB, 2012

  Let $V$ be a complex vector space with basis $\left\{e_{1}, \ldots, e_{n}\right\}$. Define $T: V \rightarrow V$ by $T\left(e_{i}\right)=e_{i}-e_{i+1}$ for $i<n$ and $T\left(e_{n}\right)=e_{n}-e_{1}$. Show that $T$ is diagonalizable and find its eigenvalues. [You may use any theorems you wish, as long as you state them clearly.]

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• Paper 4, Section II, F

  Linear Algebra | Part IB, 2012

  Let $V$ be a finite-dimensional real vector space of dimension $n$. A bilinear form $B: V \times V \rightarrow \mathbb{R}$ is nondegenerate if for all $\mathbf{v} \neq 0$ in $V$, there is some $\mathbf{w} \in V$ with $B(\mathbf{v}, \mathbf{w}) \neq 0$. For $\mathbf{v} \in V$, define $\langle\mathbf{v}\rangle^{\perp}=\{\mathbf{w} \in V \mid B(\mathbf{v}, \mathbf{w})=0\}$. Assuming $B$ is nondegenerate, show that $V=\langle\mathbf{v}\rangle \oplus\langle\mathbf{v}\rangle^{\perp}$ whenever $B(\mathbf{v}, \mathbf{v}) \neq 0$.

  Suppose that $B$ is a nondegenerate, symmetric bilinear form on $V$. Prove that there is a basis $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\}$ of $V$ with $B\left(\mathbf{v}_{i}, \mathbf{v}_{j}\right)=0$ for $i \neq j$. [If you use the fact that symmetric matrices are diagonalizable, you must prove it.]

  Define the signature of a quadratic form. Explain how to determine the signature of the quadratic form associated to $B$ from the basis you constructed above.

  A linear subspace $V^{\prime} \subset V$ is said to be isotropic if $B(\mathbf{v}, \mathbf{w})=0$ for all $\mathbf{v}, \mathbf{w} \in V^{\prime}$. Show that if $B$ is nondegenerate, the maximal dimension of an isotropic subspace of $V$ is $(n-|\sigma|) / 2$, where $\sigma$ is the signature of the quadratic form associated to $B$.

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2011

• Paper 1, Section I, G

  Linear Algebra | Part IB, 2011

  (i) State the rank-nullity theorem for a linear map between finite-dimensional vector spaces.

  (ii) Show that a linear transformation $f: V \rightarrow V$ of a finite-dimensional vector space $V$ is bijective if it is injective or surjective.

  (iii) Let $V$ be the $\mathbb{R}$-vector space $\mathbb{R}[X]$ of all polynomials in $X$ with coefficients in $\mathbb{R}$. Give an example of a linear transformation $f: V \rightarrow V$ which is surjective but not bijective.

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• Paper 1, Section II, G

  Linear Algebra | Part IB, 2011

  Let $V, W$ be finite-dimensional vector spaces over a field $F$ and $f: V \rightarrow W$ a linear map.

  (i) Show that $f$ is injective if and only if the image of every linearly independent subset of $V$ is linearly independent in $W$.

  (ii) Define the dual space $V^{*}$ of $V$ and the dual map $f^{*}: W^{*} \rightarrow V^{*}$.

  (iii) Show that $f$ is surjective if and only if the image under $f^{*}$ of every linearly independent subset of $W^{*}$ is linearly independent in $V^{*}$.

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• Paper 2, Section $I$, G

  Linear Algebra | Part IB, 2011

  Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space with an inner product. Let $W$ be an $m$-dimensional subspace of $V$ and $W^{\perp}$ its orthogonal complement, so that every element $v \in V$ can be uniquely written as $v=w+w^{\prime}$ for $w \in W$ and $w^{\prime} \in W^{\perp}$.

  The reflection map with respect to $W$ is defined as the linear map

  $$f_{W}: V \ni w+w^{\prime} \longmapsto w-w^{\prime} \in V$$

  Show that $f_{W}$ is an orthogonal transformation with respect to the inner product, and find its determinant.

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• Paper 2, Section II, G

  Linear Algebra | Part IB, 2011

  Let $n$ be a positive integer, and let $V$ be a $\mathbb{C}$-vector space of complex-valued functions on $\mathbb{R}$, generated by the set $\{\cos k x, \sin k x ; k=0,1, \ldots, n-1\}$.

  (i) Let $\langle f, g\rangle=\int_{0}^{2 \pi} f(x) \overline{g(x)} d x$ for $f, g \in V$. Show that this is a positive definite Hermitian form on $V$.

  (ii) Let $\Delta(f)=\frac{d^{2}}{d x^{2}} f(x)$. Show that $\Delta$ is a self-adjoint linear transformation of $V$ with respect to the form defined in (i).

  (iii) Find an orthonormal basis of $V$ with respect to the form defined in (i), which consists of eigenvectors of $\Delta$.

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• Paper 3, Section II, G

  Linear Algebra | Part IB, 2011

  (i) Let $A$ be an $n \times n$ complex matrix and $f(X)$ a polynomial with complex coefficients. By considering the Jordan normal form of $A$ or otherwise, show that if the eigenvalues of $A$ are $\lambda_{1}, \ldots, \lambda_{n}$ then the eigenvalues of $f(A)$ are $f\left(\lambda_{1}\right), \ldots, f\left(\lambda_{n}\right)$.

  (ii) Let $B=\left(\begin{array}{llll}a & d & c & b \\ b & a & d & c \\ c & b & a & d \\ d & c & b & a\end{array}\right)$. Write $B$ as $B=f(A)$ for a polynomial $f$ with $A=\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$, and find the eigenvalues of $B$

  [Hint: compute the powers of $A$.]

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• Paper 4, Section I, G

  Linear Algebra | Part IB, 2011

  (i) Let $V$ be a vector space over a field $F$, and $W_{1}, W_{2}$ subspaces of $V$. Define the subset $W_{1}+W_{2}$ of $V$, and show that $W_{1}+W_{2}$ and $W_{1} \cap W_{2}$ are subspaces of $V$.

  (ii) When $W_{1}, W_{2}$ are finite-dimensional, state a formula for $\operatorname{dim}\left(W_{1}+W_{2}\right)$ in terms of $\operatorname{dim} W_{1}, \operatorname{dim} W_{2}$ and $\operatorname{dim}\left(W_{1} \cap W_{2}\right)$.

  (iii) Let $V$ be the $\mathbb{R}$-vector space of all $n \times n$ matrices over $\mathbb{R}$. Let $S$ be the subspace of all symmetric matrices and $T$ the subspace of all upper triangular matrices (the matrices $\left(a_{i j}\right)$ such that $a_{i j}=0$ whenever $\left.i>j\right)$. Find $\operatorname{dim} S, \operatorname{dim} T, \operatorname{dim}(S \cap T)$ and $\operatorname{dim}(S+T)$. Briefly justify your answer.

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• Paper 4, Section II, G

  Linear Algebra | Part IB, 2011

  Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space and $f, g: V \rightarrow V$ linear transformations. Suppose $f$ is invertible and diagonalisable, and $f \circ g=t \cdot(g \circ f)$ for some real number $t>1$.

  (i) Show that $g$ is nilpotent, i.e. some positive power of $g$ is 0 .

  (ii) Suppose that there is a non-zero vector $v \in V$ with $f(v)=v$ and $g^{n-1}(v) \neq 0$. Determine the diagonal form of $f$.

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2010

• Paper 1, Section I, F

  Linear Algebra | Part IB, 2010

  Suppose that $V$ is the complex vector space of polynomials of degree at most $n-1$ in the variable $z$. Find the Jordan normal form for each of the linear transformations $\frac{d}{d z}$ and $z \frac{d}{d z}$ acting on $V$.

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• Paper 1, Section II, F

  Linear Algebra | Part IB, 2010

  Let $V$ denote the vector space of $n \times n$ real matrices.

  (1) Show that if $\psi(A, B)=\operatorname{tr}\left(A B^{T}\right)$, then $\psi$ is a positive-definite symmetric bilinear form on $V$.

  (2) Show that if $q(A)=\operatorname{tr}\left(A^{2}\right)$, then $q$ is a quadratic form on $V$. Find its rank and signature.

  [Hint: Consider symmetric and skew-symmetric matrices.]

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• Paper 2, Section I, F

  Linear Algebra | Part IB, 2010

  Suppose that $\phi$ is an endomorphism of a finite-dimensional complex vector space.

  (i) Show that if $\lambda$ is an eigenvalue of $\phi$, then $\lambda^{2}$ is an eigenvalue of $\phi^{2}$.

  (ii) Show conversely that if $\mu$ is an eigenvalue of $\phi^{2}$, then there is an eigenvalue $\lambda$ of $\phi$ with $\lambda^{2}=\mu$.

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• Paper 2, Section II, F

  Linear Algebra | Part IB, 2010

  (i) Show that two $n \times n$ complex matrices $A, B$ are similar (i.e. there exists invertible $P$ with $A=P^{-1} B P$ ) if and only if they represent the same linear map $\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ with respect to different bases.

  (ii) Explain the notion of Jordan normal form of a square complex matrix.

  (iii) Show that any square complex matrix $A$ is similar to its transpose.

  (iv) If $A$ is invertible, describe the Jordan normal form of $A^{-1}$ in terms of that of $A$.

  Justify your answers.

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• Paper 3, Section II, F

  Linear Algebra | Part IB, 2010

  Suppose that $V$ is a finite-dimensional vector space over $\mathbb{C}$, and that $\alpha: V \rightarrow V$ is a $\mathbb{C}$-linear map such that $\alpha^{n}=1$ for some $n>1$. Show that if $V_{1}$ is a subspace of $V$ such that $\alpha\left(V_{1}\right) \subset V_{1}$, then there is a subspace $V_{2}$ of $V$ such that $V=V_{1} \oplus V_{2}$ and $\alpha\left(V_{2}\right) \subset V_{2}$.

  [Hint: Show, for example by picking bases, that there is a linear map $\pi: V \rightarrow V_{1}$ with $\pi(x)=x$ for all $x \in V_{1}$. Then consider $\rho: V \rightarrow V_{1}$ with $\left.\rho(y)=\frac{1}{n} \sum_{i=0}^{n-1} \alpha^{i} \pi \alpha^{-i}(y) .\right]$

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• Paper 4, Section I, F

  Linear Algebra | Part IB, 2010

  Define the notion of an inner product on a finite-dimensional real vector space $V$, and the notion of a self-adjoint linear map $\alpha: V \rightarrow V$.

  Suppose that $V$ is the space of real polynomials of degree at most $n$ in a variable $t$. Show that

  $$\langle f, g\rangle=\int_{-1}^{1} f(t) g(t) d t$$

  is an inner product on $V$, and that the map $\alpha: V \rightarrow V$ :

  $$\alpha(f)(t)=\left(1-t^{2}\right) f^{\prime \prime}(t)-2 t f^{\prime}(t)$$

  is self-adjoint.

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• Paper 4, Section II, F

  Linear Algebra | Part IB, 2010

  (i) Show that the group $O_{n}(\mathbb{R})$ of orthogonal $n \times n$ real matrices has a normal subgroup $S O_{n}(\mathbb{R})=\left\{A \in O_{n}(\mathbb{R}) \mid \operatorname{det} A=1\right\}$.

  (ii) Show that $O_{n}(\mathbb{R})=S O_{n}(\mathbb{R}) \times\left\{\pm I_{n}\right\}$ if and only if $n$ is odd.

  (iii) Show that if $n$ is even, then $O_{n}(\mathbb{R})$ is not the direct product of $S O_{n}(\mathbb{R})$ with any normal subgroup.

  [You may assume that the only elements of $O_{n}(\mathbb{R})$ that commute with all elements of $O_{n}(\mathbb{R})$ are $\pm I_{n}$.]

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2009

• Paper 1, Section I, G

  Linear Algebra | Part IB, 2009

  (1) Let $V$ be a finite-dimensional vector space and let $T: V \rightarrow V$ be a non-zero endomorphism of $V$. If $\operatorname{ker}(T)=\operatorname{im}(T)$ show that the dimension of $V$ is an even integer. Find the minimal polynomial of $T$. [You may assume the rank-nullity theorem.]

  (2) Let $A_{i}, 1 \leqslant i \leqslant 3$, be non-zero subspaces of a vector space $V$ with the property that

  $$V=A_{1} \oplus A_{2}=A_{2} \oplus A_{3}=A_{1} \oplus A_{3} .$$

  Show that there is a 2-dimensional subspace $W \subset V$ for which all the $W \cap A_{i}$ are one-dimensional.

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• Paper 1, Section II, G

  Linear Algebra | Part IB, 2009

  Define the dual of a vector space $V$. State and prove a formula for its dimension.

  Let $V$ be the vector space of real polynomials of degree at most $n$. If $\left\{a_{0}, \ldots, a_{n}\right\}$ are distinct real numbers, prove that there are unique real numbers $\left\{\lambda_{0}, \ldots, \lambda_{n}\right\}$ with

  $$\frac{d p}{d x}(0)=\sum_{j=0}^{n} \lambda_{j} p\left(a_{j}\right)$$

  for every $p(x) \in V$.

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• Paper 2, Section I, $1 \mathbf{G}$

  Linear Algebra | Part IB, 2009

  Let $V$ denote the vector space of polynomials $f(x, y)$ in two variables of total degree at most $n$. Find the dimension of $V$.

  If $S: V \rightarrow V$ is defined by

  $$(S f)(x, y)=x^{2} \frac{\partial^{2} f}{\partial x^{2}}+y^{2} \frac{\partial^{2} f}{\partial y^{2}}$$

  find the kernel of $S$ and the image of $S$. Compute the trace of $S$ for each $n$ with $1 \leqslant n \leqslant 4$.

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• Paper 2, Section II, G

  Linear Algebra | Part IB, 2009

  Let $V$ be a finite-dimensional vector space and let $T: V \rightarrow V$ be an endomorphism of $V$. Show that there is a positive integer $l$ such that $V=\operatorname{ker}\left(T^{l}\right) \oplus \operatorname{im}\left(T^{l}\right)$. Hence, or otherwise, show that if $T$ has zero determinant there is some non-zero endomorphism $S$ with $T S=0=S T$.

  Suppose $T_{1}$ and $T_{2}$ are endomorphisms of $V$ for which $T_{i}^{2}=T_{i}, i=1,2$. Show that $T_{1}$ is similar to $T_{2}$ if and only if they have the same rank.

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• Paper 3, Section II, G

  Linear Algebra | Part IB, 2009

  For each of the following, provide a proof or counterexample.

  (1) If $A, B$ are complex $n \times n$ matrices and $A B=B A$, then $A$ and $B$ have a common eigenvector.

  (2) If $A, B$ are complex $n \times n$ matrices and $A B=B A$, then $A$ and $B$ have a common eigenvalue.

  (3) If $A, B$ are complex $n \times n$ matrices and $(A B)^{n}=0$ then $(B A)^{n}=0$.

  (4) If $T: V \rightarrow V$ is an endomorphism of a finite-dimensional vector space $V$ and $\lambda$ is an eigenvalue of $T$, then the dimension of $\{v \in V \mid(T-\lambda I) v=0\}$ equals the multiplicity of $\lambda$ as a root of the minimal polynomial of $T$.

  (5) If $T: V \rightarrow V$ is an endomorphism of a finite-dimensional complex vector space $V$, $\lambda$ is an eigenvalue of $T$, and $W_{i}=\left\{v \in V \mid(T-\lambda I)^{i}(v)=0\right\}$, then $W_{c}=W_{c+1}$ where $c$ is the multiplicity of $\lambda$ as a root of the minimal polynomial of $T$.

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• Paper 4, Section I, G

  Linear Algebra | Part IB, 2009

  Show that every endomorphism of a finite-dimensional vector space satisfies some polynomial, and define the minimal polynomial of such an endomorphism.

  Give a linear transformation of an eight-dimensional complex vector space which has minimal polynomial $x^{2}(x-1)^{3}$.

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• Paper 4, Section II, G

  Linear Algebra | Part IB, 2009

  What does it mean to say two real symmetric bilinear forms $A$ and $B$ on a vector space $V$ are congruent ?

  State and prove Sylvester's law of inertia, and deduce that the rank and signature determine the congruence class of a real symmetric bilinear form. [You may use without proof a result on diagonalisability of real symmetric matrices, provided it is clearly stated.]

  How many congruence classes of symmetric bilinear forms on a real $n$-dimensional vector space are there? Such a form $\psi$ defines a family of subsets $\left\{x \in \mathbb{R}^{n} \mid \psi(x, x)=t\right\}$, for $t \in \mathbb{R}$. For how many of the congruence classes are these associated subsets all bounded subsets of $\mathbb{R}^{n}$ ? Is the quadric surface

  $$\left\{3 x^{2}+6 y^{2}+5 z^{2}+4 x y+2 x z+8 y z=1\right\}$$

  a bounded or unbounded subset of $\mathbb{R}^{3}$ ? Justify your answers.

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2008

• 1.I.1E

  Linear Algebra | Part IB, 2008

  Let $A$ be an $n \times n$ matrix over $\mathbb{C}$. What does it mean to say that $\lambda$ is an eigenvalue of $A$ ? Show that $A$ has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.

  (i) If $A$ is Hermitian, all eigenvalues of $A$ are real.

  (ii) If all eigenvalues of $A$ are real, $A$ is Hermitian.

  (iii) If all entries of $A$ are real and positive, all eigenvalues of $A$ have positive real part.

  (iv) If $A$ and $B$ have the same trace and determinant then they have the same eigenvalues.

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• 1.II.9E

  Linear Algebra | Part IB, 2008

  Let $A$ be an $m \times n$ matrix of real numbers. Define the row rank and column rank of $A$ and show that they are equal.

  Show that if a matrix $A^{\prime}$ is obtained from $A$ by elementary row and column operations then $\operatorname{rank}\left(A^{\prime}\right)=\operatorname{rank}(A)$.

  Let $P, Q$ and $R$ be $n \times n$ matrices. Show that the $2 n \times 2 n$ matrices $\left(\begin{array}{cc}P Q & 0 \\ Q & Q R\end{array}\right)$ and $\left(\begin{array}{cc}0 & P Q R \\ Q & 0\end{array}\right)$ have the same rank.

  Hence, or otherwise, prove that

  $$\operatorname{rank}(P Q)+\operatorname{rank}(Q R) \leqslant \operatorname{rank}(Q)+\operatorname{rank}(P Q R)$$

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• 2.I.1E

  Linear Algebra | Part IB, 2008

  Suppose that $V$ and $W$ are finite-dimensional vector spaces over $\mathbb{R}$. What does it mean to say that $\psi: V \rightarrow W$ is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map $\psi: V \rightarrow V$ is surjective if, and only if, it is injective.

  Suppose that $\psi: V \rightarrow V$ is a linear map which has a right inverse, that is to say there is a linear map $\phi: V \rightarrow V$ such that $\psi \phi=\mathrm{id}_{V}$, the identity map. Show that $\phi \psi=\mathrm{id}_{V}$.

  Suppose that $A$ and $B$ are two $n \times n$ matrices over $\mathbb{R}$ such that $A B=I$. Prove that $B A=I$.

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• 2.II.10E

  Linear Algebra | Part IB, 2008

  Define the determinant $\operatorname{det}(A)$ of an $n \times n$ square matrix $A$ over the complex numbers. If $A$ and $B$ are two such matrices, show that $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$.

  Write $p_{M}(\lambda)=\operatorname{det}(M-\lambda I)$ for the characteristic polynomial of a matrix $M$. Let $A, B, C$ be $n \times n$ matrices and suppose that $C$ is nonsingular. Show that $p_{B C}=p_{C B}$. Taking $C=A+t I$ for appropriate values of $t$, or otherwise, deduce that $p_{B A}=p_{A B}$.

  Show that if $p_{A}=p_{B}$ then $\operatorname{tr}(A)=\operatorname{tr}(B)$. Which of the following statements is true for all $n \times n$ matrices $A, B, C$ ? Justify your answers.

  (i) $p_{A B C}=p_{A C B}$;

  (ii) $p_{A B C}=p_{B C A}$.

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• 3.II.10E

  Linear Algebra | Part IB, 2008

  Let $k=\mathbb{R}$ or $\mathbb{C}$. What is meant by a quadratic form $q: k^{n} \rightarrow k$ ? Show that there is a basis $\left\{v_{1}, \ldots, v_{n}\right\}$ for $k^{n}$ such that, writing $x=x_{1} v_{1}+\ldots+x_{n} v_{n}$, we have $q(x)=a_{1} x_{1}^{2}+\ldots+a_{n} x_{n}^{2}$ for some scalars $a_{1}, \ldots, a_{n} \in\{-1,0,1\} .$

  Suppose that $k=\mathbb{R}$. Define the rank and signature of $q$ and compute these quantities for the form $q: \mathbb{R}^{3} \rightarrow \mathbb{R}$ given by $q(x)=-3 x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}$.

  Suppose now that $k=\mathbb{C}$ and that $q_{1}, \ldots, q_{d}: \mathbb{C}^{n} \rightarrow \mathbb{C}$ are quadratic forms. If $n \geqslant 2^{d}$, show that there is some nonzero $x \in \mathbb{C}^{n}$ such that $q_{1}(x)=\ldots=q_{d}(x)=0$.

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• 4.I.1E

  Linear Algebra | Part IB, 2008

  Describe (without proof) what it means to put an $n \times n$ matrix of complex numbers into Jordan normal form. Explain (without proof) the sense in which the Jordan normal form is unique.

  Put the following matrix in Jordan normal form:

  $$\left(\begin{array}{ccc} -7 & 3 & -5 \\ 7 & -1 & 5 \\ 17 & -6 & 12 \end{array}\right)$$

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• 4.II.10E

  Linear Algebra | Part IB, 2008

  What is meant by a Hermitian matrix? Show that if $A$ is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for $\mathbb{C}^{n}$ consisting of eigenvectors of $A$.

  A Hermitian matrix is said to be positive definite if $\langle A x, x\rangle>0$ for all $x \neq 0$. We write $A>0$ in this case. Show that $A$ is positive definite if, and only if, all of its eigenvalues are positive. Show that if $A>0$ then $A$ has a unique positive definite square root $\sqrt{A}$.

  Let $A, B$ be two positive definite Hermitian matrices with $A-B>0$. Writing $C=\sqrt{A}$ and $X=\sqrt{A}-\sqrt{B}$, show that $C X+X C>0$. By considering eigenvalues of $X$, or otherwise, show that $X>0$.

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2007

• 1.I.1G

  Linear Algebra | Part IB, 2007

  Suppose that $\left\{e_{1}, \ldots, e_{3}\right\}$ is a basis of the complex vector space $\mathbb{C}^{3}$ and that $A: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ is the linear operator defined by $A\left(e_{1}\right)=e_{2}, A\left(e_{2}\right)=e_{3}$, and $A\left(e_{3}\right)=e_{1}$.

  By considering the action of $A$ on column vectors of the form $\left(1, \xi, \xi^{2}\right)^{T}$, where $\xi^{3}=1$, or otherwise, find the diagonalization of $A$ and its characteristic polynomial.

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• 1.II.9G

  Linear Algebra | Part IB, 2007

  State and prove Sylvester's law of inertia for a real quadratic form.

  [You may assume that for each real symmetric matrix A there is an orthogonal matrix $U$, such that $U^{-1} A U$ is diagonal.]

  Suppose that $V$ is a real vector space of even dimension $2 m$, that $Q$ is a non-singular quadratic form on $V$ and that $U$ is an $m$-dimensional subspace of $V$ on which $Q$ vanishes. What is the signature of $Q ?$

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• 2.I.1G

  Linear Algebra | Part IB, 2007

  Suppose that $S, T$ are endomorphisms of the 3-dimensional complex vector space $\mathbb{C}^{3}$ and that the eigenvalues of each of them are $1,2,3$. What are their characteristic and minimal polynomials? Are they conjugate?

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• 2.II.10G

  Linear Algebra | Part IB, 2007

  Suppose that $P$ is the complex vector space of complex polynomials in one variable, $z$.

  (i) Show that the form $\langle$, $\rangle$ defined by

  $$\langle f, g\rangle=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(e^{i \theta}\right) \cdot \overline{g\left(e^{i \theta}\right)} d \theta$$

  is a positive definite Hermitian form on $P$.

  (ii) Find an orthonormal basis of $P$ for this form, in terms of the powers of $z$.

  (iii) Generalize this construction to complex vector spaces of complex polynomials in any finite number of variables.

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• 3.II.10G

  Linear Algebra | Part IB, 2007

  (i) Define the terms row-rank, column-rank and rank of a matrix, and state a relation between them.

  (ii) Fix positive integers $m, n, p$ with $m, n \geqslant p$. Suppose that $A$ is an $m \times p$ matrix and $B$ a $p \times n$ matrix. State and prove the best possible upper bound on the rank of the product $A B$.

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• 4.I.1G

  Linear Algebra | Part IB, 2007

  Suppose that $\alpha: V \rightarrow W$ is a linear map of finite-dimensional complex vector spaces. What is the dual map $\alpha^{*}$ of the dual vector spaces?

  Suppose that we choose bases of $V, W$ and take the corresponding dual bases of the dual vector spaces. What is the relation between the matrices that represent $\alpha$ and $\alpha^{*}$ with respect to these bases? Justify your answer.

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• 4.II.10G

  Linear Algebra | Part IB, 2007

  (i) State and prove the Cayley-Hamilton theorem for square complex matrices.

  (ii) A square matrix $A$ is of order $n$ for a strictly positive integer $n$ if $A^{n}=I$ and no smaller positive power of $A$ is equal to $I$.

  Determine the order of a complex $2 \times 2$ matrix $A$ of trace zero and determinant 1 .

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2006

• 1.I.1H

  Linear Algebra | Part IB, 2006

  Define what is meant by the minimal polynomial of a complex $n \times n$ matrix, and show that it is unique. Deduce that the minimal polynomial of a real $n \times n$ matrix has real coefficients.

  For $n>2$, find an $n \times n$ matrix with minimal polynomial $(t-1)^{2}(t+1)$.

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• 1.II.9H

  Linear Algebra | Part IB, 2006

  Let $U, V$ be finite-dimensional vector spaces, and let $\theta$ be a linear map of $U$ into $V$. Define the rank $r(\theta)$ and the nullity $n(\theta)$ of $\theta$, and prove that

  $$r(\theta)+n(\theta)=\operatorname{dim} U$$

  Now let $\theta, \phi$ be endomorphisms of a vector space $U$. Define the endomorphisms $\theta+\phi$ and $\theta \phi$, and prove that

  $$\begin{aligned} r(\theta+\phi) & \leqslant r(\theta)+r(\phi) \\ n(\theta \phi) & \leqslant n(\theta)+n(\phi) . \end{aligned}$$

  Prove that equality holds in both inequalities if and only if $\theta+\phi$ is an isomorphism and $\theta \phi$ is zero.

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• 2.I.1E

  Linear Algebra | Part IB, 2006

  State Sylvester's law of inertia.

  Find the rank and signature of the quadratic form $q$ on $\mathbf{R}^{n}$ given by

  $$q\left(x_{1}, \ldots, x_{n}\right)=\left(\sum_{i=1}^{n} x_{i}\right)^{2}-\sum_{i=1}^{n} x_{i}^{2}$$

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• 2.II.10E

  Linear Algebra | Part IB, 2006

  Suppose that $V$ is the set of complex polynomials of degree at most $n$ in the variable $x$. Find the dimension of $V$ as a complex vector space.

  Define

  $$e_{k}: V \rightarrow \mathbf{C} \quad \text { by } \quad e_{k}(\phi)=\frac{d^{k} \phi}{d x^{k}}(0)$$

  Find a subset of $\left\{e_{k} \mid k \in \mathbf{N}\right\}$ that is a basis of the dual vector space $V^{*}$. Find the corresponding dual basis of $V$.

  Define

  $$D: V \rightarrow V \quad \text { by } \quad D(\phi)=\frac{d \phi}{d x} .$$

  Write down the matrix of $D$ with respect to the basis of $V$ that you have just found, and the matrix of the map dual to $D$ with respect to the dual basis.

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• 3.II.10H

  Linear Algebra | Part IB, 2006

  (a) Define what is meant by the trace of a complex $n \times n$ matrix $A$. If $T$ denotes an $n \times n$ invertible matrix, show that $A$ and $T A T^{-1}$ have the same trace.

  (b) If $\lambda_{1}, \ldots, \lambda_{r}$ are distinct non-zero complex numbers, show that the endomorphism of $\mathbf{C}^{r}$ defined by the matrix

  $$\Lambda=\left(\begin{array}{ccc} \lambda_{1} & \ldots & \lambda_{1}^{r} \\ \vdots & \ldots & \vdots \\ \lambda_{r} & \ldots & \lambda_{r}^{r} \end{array}\right)$$

  has trivial kernel, and hence that the same is true for the transposed matrix $\Lambda^{t}$.

  For arbitrary complex numbers $\lambda_{1}, \ldots, \lambda_{n}$, show that the vector $(1, \ldots, 1)^{t}$ is not in the kernel of the endomorphism of $\mathbf{C}^{n}$ defined by the matrix

  $$\left(\begin{array}{ccc} \lambda_{1} & \ldots & \lambda_{n} \\ \vdots & \ldots & \vdots \\ \lambda_{1}^{n} & \ldots & \lambda_{n}^{n} \end{array}\right)$$

  unless all the $\lambda_{i}$ are zero.

  [Hint: reduce to the case when $\lambda_{1}, \ldots, \lambda_{r}$ are distinct non-zero complex numbers, with $r \leqslant n$, and each $\lambda_{j}$ for $j>r$ is either zero or equal to some $\lambda_{i}$ with $i \leqslant r$. If the kernel of the endomorphism contains $(1, \ldots, 1)^{t}$, show that it also contains a vector of the form $\left(m_{1}, \ldots, m_{r}, 0, \ldots, 0\right)^{t}$ with the $m_{i}$ strictly positive integers.]

  (c) Assuming the fact that any complex $n \times n$ matrix is conjugate to an uppertriangular one, prove that if $A$ is an $n \times n$ matrix such that $A^{k}$ has zero trace for all $1 \leqslant k \leqslant n$, then $A^{n}=0 .$

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• 4.I.1H

  Linear Algebra | Part IB, 2006

  Suppose $V$ is a vector space over a field $k$. A finite set of vectors is said to be a basis for $V$ if it is both linearly independent and spanning. Prove that any two finite bases for $V$ have the same number of elements.

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• 4.II.10E

  Linear Algebra | Part IB, 2006

  Suppose that $\alpha$ is an orthogonal endomorphism of the finite-dimensional real inner product space $V$. Suppose that $V$ is decomposed as a direct sum of mutually orthogonal $\alpha$-invariant subspaces. How small can these subspaces be made, and how does $\alpha$ act on them? Justify your answer.

  Describe the possible matrices for $\alpha$ with respect to a suitably chosen orthonormal basis of $V$ when $\operatorname{dim} V=3$.

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2005

• 1.I.1C

  Linear Algebra | Part IB, 2005

  Let $V$ be an $n$-dimensional vector space over $\mathbf{R}$, and let $\beta: V \rightarrow V$ be a linear map. Define the minimal polynomial of $\beta$. Prove that $\beta$ is invertible if and only if the constant term of the minimal polynomial of $\beta$ is non-zero.

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• 1.II.9C

  Linear Algebra | Part IB, 2005

  Let $V$ be a finite dimensional vector space over $\mathbf{R}$, and $V^{*}$ be the dual space of $V$.

  If $W$ is a subspace of $V$, we define the subspace $\alpha(W)$ of $V^{*}$ by

  $$\alpha(W)=\left\{f \in V^{*}: f(w)=0 \text { for all } w \text { in } W\right\}$$

  Prove that $\operatorname{dim}(\alpha(W))=\operatorname{dim}(V)-\operatorname{dim}(W)$. Deduce that, if $A=\left(a_{i j}\right)$ is any real $m \times n$-matrix of rank $r$, the equations

  $$\sum_{j=1}^{n} a_{i j} x_{j}=0 \quad(i=1, \ldots, m)$$

  have $n-r$ linearly independent solutions in $\mathbf{R}^{n}$.

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• 2.I.1C

  Linear Algebra | Part IB, 2005

  Let $\Omega$ be the set of all $2 \times 2$ matrices of the form $\alpha=a I+b J+c K+d L$, where $a, b, c, d$ are in $\mathbf{R}$, and

  $$I=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right), J=\left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right), K=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), L=\left(\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right) \quad\left(i^{2}=-1\right) .$$

  Prove that $\Omega$ is closed under multiplication and determine its dimension as a vector space over $\mathbf{R}$. Prove that

  $$(a I+b J+c K+d L)(a I-b J-c K-d L)=\left(a^{2}+b^{2}+c^{2}+d^{2}\right) I$$

  and deduce that each non-zero element of $\Omega$ is invertible.

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• 2.II.10C

  Linear Algebra | Part IB, 2005

  (i) Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix with entries in C. Define the determinant of $A$, the cofactor of each $a_{i j}$, and the adjugate matrix $\operatorname{adj}(A)$. Assuming the expansion of the determinant of a matrix in terms of its cofactors, prove that

  $$\operatorname{adj}(A) A=\operatorname{det}(A) I_{n}$$

  where $I_{n}$ is the $n \times n$ identity matrix.

  (ii) Let

  $$A=\left(\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right)$$

  Show the eigenvalues of $A$ are $\pm 1, \pm i$, where $i^{2}=-1$, and determine the diagonal matrix to which $A$ is similar. For each eigenvalue, determine a non-zero eigenvector.

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• 3.II.10B

  Linear Algebra | Part IB, 2005

  Let $S$ be the vector space of functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that the $n$th derivative of $f$ is defined and continuous for every $n \geqslant 0$. Define linear maps $A, B: S \rightarrow S$ by $A(f)=d f / d x$ and $B(f)(x)=x f(x)$. Show that

  $$[A, B]=1_{S},$$

  where in this question $[A, B]$ means $A B-B A$ and $1_{S}$ is the identity map on $S$.

  Now let $V$ be any real vector space with linear maps $A, B: V \rightarrow V$ such that $[A, B]=1_{V}$. Suppose that there is a nonzero element $y \in V$ with $A y=0$. Let $W$ be the subspace of $V$ spanned by $y, B y, B^{2} y$, and so on. Show that $A(B y)$ is in $W$ and give a formula for it. More generally, show that $A\left(B^{i} y\right)$ is in $W$ for each $i \geqslant 0$, and give a formula for it.

  Show, using your formula or otherwise, that $\left\{y, B y, B^{2} y, \ldots\right\}$ are linearly independent. (Or, equivalently: show that $y, B y, B^{2} y, \ldots, B^{n} y$ are linearly independent for every $n \geqslant 0$.)

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• 4.I.1B

  Linear Algebra | Part IB, 2005

  Define what it means for an $n \times n$ complex matrix to be unitary or Hermitian. Show that every eigenvalue of a Hermitian matrix is real. Show that every eigenvalue of a unitary matrix has absolute value 1 .

  Show that two eigenvectors of a Hermitian matrix that correspond to different eigenvalues are orthogonal, using the standard inner product on $\mathbf{C}^{n}$.

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• 4.II.10B

  Linear Algebra | Part IB, 2005

  (i) Let $V$ be a finite-dimensional real vector space with an inner product. Let $e_{1}, \ldots, e_{n}$ be a basis for $V$. Prove by an explicit construction that there is an orthonormal basis $f_{1}, \ldots, f_{n}$ for $V$ such that the span of $e_{1}, \ldots, e_{i}$ is equal to the span of $f_{1}, \ldots, f_{i}$ for every $1 \leqslant i \leqslant n$.

  (ii) For any real number $a$, consider the quadratic form

  $$q_{a}(x, y, z)=x y+y z+z x+a x^{2}$$

  on $\mathbf{R}^{3}$. For which values of $a$ is $q_{a}$ nondegenerate? When $q_{a}$ is nondegenerate, compute its signature in terms of $a$.

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2004

• 1.I.1H

  Linear Algebra | Part IB, 2004

  Suppose that $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r+1}\right\}$ is a linearly independent set of distinct elements of a vector space $V$ and $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r}, \mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$. Prove that $\mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}$ may be reordered, as necessary, so that $\left\{\mathbf{e}_{1}, \ldots \mathbf{e}_{r+1}, \mathbf{f}_{r+2}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$.

  Suppose that $\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\}$ is a linearly independent set of distinct elements of $V$ and that $\left\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{m}\right\}$ spans $V$. Show that $n \leqslant m$.

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• 1.II.12H

  Linear Algebra | Part IB, 2004

  Let $U$ and $W$ be subspaces of the finite-dimensional vector space $V$. Prove that both the sum $U+W$ and the intersection $U \cap W$ are subspaces of $V$. Prove further that

  $$\operatorname{dim} U+\operatorname{dim} W=\operatorname{dim}(U+W)+\operatorname{dim}(U \cap W)$$

  Let $U, W$ be the kernels of the maps $A, B: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}$ given by the matrices $A$ and $B$ respectively, where

  $$A=\left(\begin{array}{rrrr} 1 & 2 & -1 & -3 \\ -1 & 1 & 2 & -4 \end{array}\right), \quad B=\left(\begin{array}{rrrr} 1 & -1 & 2 & 0 \\ 0 & 1 & 2 & -4 \end{array}\right)$$

  Find a basis for the intersection $U \cap W$, and extend this first to a basis of $U$, and then to a basis of $U+W$.

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• 2.I.1E

  Linear Algebra | Part IB, 2004

  For each $n$ let $A_{n}$ be the $n \times n$ matrix defined by

  $$\left(A_{n}\right)_{i j}= \begin{cases}i & i \leqslant j \\ j & i>j\end{cases}$$

  What is $\operatorname{det} A_{n} ?$ Justify your answer.

  [It may be helpful to look at the cases $n=1,2,3$ before tackling the general case.]

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• 2.II.12E

  Linear Algebra | Part IB, 2004

  Let $Q$ be a quadratic form on a real vector space $V$ of dimension $n$. Prove that there is a basis $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ with respect to which $Q$ is given by the formula

  $$Q\left(\sum_{i=1}^{n} x_{i} \mathbf{e}_{i}\right)=x_{1}^{2}+\ldots+x_{p}^{2}-x_{p+1}^{2}-\ldots-x_{p+q}^{2}$$

  Prove that the numbers $p$ and $q$ are uniquely determined by the form $Q$. By means of an example, show that the subspaces $\left\langle\mathbf{e}_{1}, \ldots, \mathbf{e}_{p}\right\rangle$ and $\left\langle\mathbf{e}_{p+1}, \ldots, \mathbf{e}_{p+q}\right\rangle$ need not be uniquely determined by $Q$.

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• 3.I.1E

  Linear Algebra | Part IB, 2004

  Let $V$ be a finite-dimensional vector space over $\mathbb{R}$. What is the dual space of $V$ ? Prove that the dimension of the dual space is the same as that of $V$.

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• 3.II.13E

  Linear Algebra | Part IB, 2004

  (i) Let $V$ be an $n$-dimensional vector space over $\mathbb{C}$ and let $\alpha: V \rightarrow V$ be an endomorphism. Suppose that the characteristic polynomial of $\alpha$ is $\Pi_{i=1}^{k}\left(x-\lambda_{i}\right)^{n_{i}}$, where the $\lambda_{i}$ are distinct and $n_{i}>0$ for every $i$.

  Describe all possibilities for the minimal polynomial and prove that there are no further ones.

  (ii) Give an example of a matrix for which both the characteristic and the minimal polynomial are $(x-1)^{3}(x-3)$.

  (iii) Give an example of two matrices $A, B$ with the same rank and the same minimal and characteristic polynomials such that there is no invertible matrix $P$ with $P A P^{-1}=B$.

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• 4.I.1E

  Linear Algebra | Part IB, 2004

  Let $V$ be a real $n$-dimensional inner-product space and let $W \subset V$ be a $k$ dimensional subspace. Let $\mathbf{e}_{1}, \ldots, \mathbf{e}_{k}$ be an orthonormal basis for $W$. In terms of this basis, give a formula for the orthogonal projection $\pi: V \rightarrow W$.

  Let $v \in V$. Prove that $\pi v$ is the closest point in $W$ to $v$.

  [You may assume that the sequence $\mathbf{e}_{1}, \ldots, \mathbf{e}_{k}$ can be extended to an orthonormal basis $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ of $V$.]

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• 4.II.11E

  Linear Algebra | Part IB, 2004

  (i) Let $V$ be an $n$-dimensional inner-product space over $\mathbb{C}$ and let $\alpha: V \rightarrow V$ be a Hermitian linear map. Prove that $V$ has an orthonormal basis consisting of eigenvectors of $\alpha$.

  (ii) Let $\beta: V \rightarrow V$ be another Hermitian map. Prove that $\alpha \beta$ is Hermitian if and only if $\alpha \beta=\beta \alpha$.

  (iii) A Hermitian map $\alpha$ is positive-definite if $\langle\alpha v, v\rangle>0$ for every non-zero vector $v$. If $\alpha$ is a positive-definite Hermitian map, prove that there is a unique positivedefinite Hermitian map $\beta$ such that $\beta^{2}=\alpha$.

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