• # 1.I.8G

Let $U$ and $V$ be finite-dimensional vector spaces. Suppose that $b$ and $c$ are bilinear forms on $U \times V$ and that $b$ is non-degenerate. Show that there exist linear endomorphisms $S$ of $U$ and $T$ of $V$ such that $c(x, y)=b(S(x), y)=b(x, T(y))$ for all $(x, y) \in U \times V$.

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• # 1.II.17G

(a) Suppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$ and state Euler's criterion.

(b) Compute $\left(\frac{-1}{p}\right)$ and prove that

$\left(\frac{a b}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$

whenever $a$ and $b$ are coprime to $p$.

(c) Let $n$ be any integer such that $1 \leqslant n \leqslant p-2$. Let $m$ be the unique integer such that $1 \leqslant m \leqslant p-2$ and $m n \equiv 1(\bmod p)$. Prove that

$\left(\frac{n(n+1)}{p}\right)=\left(\frac{1+m}{p}\right)$

(d) Find

$\sum_{n=1}^{p-2}\left(\frac{n(n+1)}{p}\right)$

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• # 2.I.8G

Let $U$ be a finite-dimensional real vector space and $b$ a positive definite symmetric bilinear form on $U \times U$. Let $\psi: U \rightarrow U$ be a linear map such that $b(\psi(x), y)+b(x, \psi(y))=0$ for all $x$ and $y$ in $U$. Prove that if $\psi$ is invertible, then the dimension of $U$ must be even. By considering the restriction of $\psi$ to its image or otherwise, prove that the rank of $\psi$ is always even.

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• # 2.II.17G

Let $S$ be the set of all $2 \times 2$ complex matrices $A$ which are hermitian, that is, $A^{*}=A$, where $A^{*}=\bar{A}^{t}$.

(a) Show that $S$ is a real 4-dimensional vector space. Consider the real symmetric bilinear form $b$ on this space defined by

$b(A, B)=\frac{1}{2}(\operatorname{tr}(A B)-\operatorname{tr}(A) \operatorname{tr}(B)) .$

Prove that $b(A, A)=-\operatorname{det} A$ and $b(A, I)=-\frac{1}{2} \operatorname{tr}(A)$, where $I$ denotes the identity matrix.

(b) Consider the three matrices

$A_{1}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), \quad A_{2}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \text { and } \quad A_{3}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)$

Prove that the basis $I, A_{1}, A_{2}, A_{3}$ of $S$ diagonalizes $b$. Hence or otherwise find the rank and signature of $b$.

(c) Let $Q$ be the set of all $2 \times 2$ complex matrices $C$ which satisfy $C+C^{*}=\operatorname{tr}(C) I$. Show that $Q$ is a real 4-dimensional vector space. Given $C \in Q$, put

$\Phi(C)=\frac{1-i}{2} \operatorname{tr}(C) I+i C .$

Show that $\Phi$ takes values in $S$ and is a linear isomorphism between $Q$ and $S$.

(d) Define a real symmetric bilinear form on $Q$ by setting $c(C, D)=-\frac{1}{2} \operatorname{tr}(C D)$, $C, D \in Q$. Show that $b(\Phi(C), \Phi(D))=c(C, D)$ for all $C, D \in Q$. Find the rank and signature of the symmetric bilinear form $c$ defined on $Q$.

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• # 3.I.9G

Let $f(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Explain what is meant by the discriminant $d$ of $f$. State a necessary and sufficient condition for some form of discriminant $d$ to represent an odd prime number $p$. Using this result or otherwise, find the primes $p$ which can be represented by the form $x^{2}+3 y^{2}$.

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• # 3.II.19G

Let $U$ be a finite-dimensional real vector space endowed with a positive definite inner product. A linear map $\tau: U \rightarrow U$ is said to be an orthogonal projection if $\tau$ is self-adjoint and $\tau^{2}=\tau$.

(a) Prove that for every orthogonal projection $\tau$ there is an orthogonal decomposition

$U=\operatorname{ker}(\tau) \oplus \operatorname{im}(\tau)$

(b) Let $\phi: U \rightarrow U$ be a linear map. Show that if $\phi^{2}=\phi$ and $\phi \phi^{*}=\phi^{*} \phi$, where $\phi^{*}$ is the adjoint of $\phi$, then $\phi$ is an orthogonal projection. [You may find it useful to prove first that if $\phi \phi^{*}=\phi^{*} \phi$, then $\phi$ and $\phi^{*}$ have the same kernel.]

(c) Show that given a subspace $W$ of $U$ there exists a unique orthogonal projection $\tau$ such that $\operatorname{im}(\tau)=W$. If $W_{1}$ and $W_{2}$ are two subspaces with corresponding orthogonal projections $\tau_{1}$ and $\tau_{2}$, show that $\tau_{2} \circ \tau_{1}=0$ if and only if $W_{1}$ is orthogonal to $W_{2}$.

(d) Let $\phi: U \rightarrow U$ be a linear map satisfying $\phi^{2}=\phi$. Prove that one can define a positive definite inner product on $U$ such that $\phi$ becomes an orthogonal projection.

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• # 1.I.8F

Define the rank and signature of a symmetric bilinear form $\phi$ on a finite-dimensional real vector space. (If your definitions involve a matrix representation of $\phi$, you should explain why they are independent of the choice of representing matrix.)

Let $V$ be the space of all $n \times n$ real matrices (where $n \geqslant 2$ ), and let $\phi$ be the bilinear form on $V$ defined by

$\phi(A, B)=\operatorname{tr} A B-\operatorname{tr} A \operatorname{tr} B$

Find the rank and signature of $\phi$.

[Hint: You may find it helpful to consider the subspace of symmetric matrices having trace zero, and a suitable complement for this subspace.]

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• # 1.II.17F

Let $A$ and $B$ be $n \times n$ real symmetric matrices, such that the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ is positive definite. Show that it is possible to find an invertible matrix $P$ such that $P^{T} A P=I$ and $P^{T} B P$ is diagonal. Show also that the diagonal entries of the matrix $P^{T} B P$ may be calculated directly from $A$ and $B$, without finding the matrix $P$. If

$A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc} 4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right)$

find the diagonal entries of $P^{T} B P$.

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• # 2.I.8F

Explain what is meant by a sesquilinear form on a complex vector space $V$. If $\phi$ and $\psi$ are two such forms, and $\phi(v, v)=\psi(v, v)$ for all $v \in V$, prove that $\phi(v, w)=\psi(v, w)$ for all $v, w \in V$. Deduce that if $\alpha: V \rightarrow V$ is a linear map satisfying $\phi(\alpha(v), \alpha(v))=\phi(v, v)$ for all $v \in V$, then $\phi(\alpha(v), \alpha(w))=\phi(v, w)$ for all $v, w \in V$.

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• # 2.II.17F

Define the adjoint $\alpha^{*}$ of an endomorphism $\alpha$ of a complex inner-product space $V$. Show that if $W$ is a subspace of $V$, then $\alpha(W) \subseteq W$ if and only if $\alpha^{*}\left(W^{\perp}\right) \subseteq W^{\perp}$.

An endomorphism of a complex inner-product space is said to be normal if it commutes with its adjoint. Prove the following facts about a normal endomorphism $\alpha$ of a finite-dimensional space $V$.

(i) $\alpha$ and $\alpha^{*}$ have the same kernel.

(ii) $\alpha$ and $\alpha^{*}$ have the same eigenvectors, with complex conjugate eigenvalues.

(iii) If $E_{\lambda}=\{x \in V: \alpha(x)=\lambda x\}$, then $\alpha\left(E_{\lambda}^{\perp}\right) \subseteq E_{\lambda}^{\perp}$.

(iv) There is an orthonormal basis of $V$ consisting of eigenvectors of $\alpha$.

Deduce that an endomorphism $\alpha$ is normal if and only if it can be written as a product $\beta \gamma$, where $\beta$ is Hermitian, $\gamma$ is unitary and $\beta$ and $\gamma$ commute with each other. [Hint: Given $\alpha$, define $\beta$ and $\gamma$ in terms of their effect on the basis constructed in (iv).]

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• # 3.I $. 9 \mathrm{~F} \quad$

Explain what is meant by a quadratic residue modulo an odd prime $p$, and show that $a$ is a quadratic residue modulo $p$ if and only if $a^{\frac{1}{2}(p-1)} \equiv 1(\bmod p)$. Hence characterize the odd primes $p$ for which $-1$ is a quadratic residue.

State the law of quadratic reciprocity, and use it to determine whether 73 is a quadratic residue (mod 127).

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• # 3.II.19F

Explain what is meant by saying that a positive definite integral quadratic form $f(x, y)=a x^{2}+b x y+c y^{2}$ is reduced, and show that every positive definite form is equivalent to a reduced form

State a criterion for a prime number $p$ to be representable by some form of discriminant $d$, and deduce that $p$ is representable by a form of discriminant $-32$ if and only if $p \equiv 1,2$ or $3(\bmod 8)$. Find the reduced forms of discriminant $-32$, and hence or otherwise show that a prime $p$ is representable by the form $3 x^{2}+2 x y+3 y^{2}$ if and only if $p \equiv 3(\bmod 8)$.

[Standard results on when $-1$ and 2 are squares (mod $p$ ) may be assumed.]

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• # 1.I.8B

Let $q(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Define what is meant by the discriminant $d$ of $q$, and show that $q$ is positive-definite if and only if $a>0>d$. Define what it means for the form $q$ to be reduced. For any integer $d<0$, we define the class number $h(d)$ to be the number of positive-definite reduced binary quadratic forms (with integer coefficients) with discriminant $d$. Show that $h(d)$ is always finite (for negative $d)$. Find $h(-39)$, and exhibit the corresponding reduced forms.

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• # 1.II.17B

Let $\phi$ be a symmetric bilinear form on a finite dimensional vector space $V$ over a field $k$ of characteristic $\neq 2$. Prove that the form $\phi$ may be diagonalized, and interpret the rank $r$ of $\phi$ in terms of the resulting diagonal form.

For $\phi$ a symmetric bilinear form on a real vector space $V$ of finite dimension $n$, define the signature $\sigma$ of $\phi$, proving that it is well-defined. A subspace $U$ of $V$ is called null if $\left.\phi\right|_{U} \equiv 0$; show that $V$ has a null subspace of dimension $n-\frac{1}{2}(r+|\sigma|)$, but no null subspace of higher dimension.

Consider now the quadratic form $q$ on $\mathbb{R}^{5}$ given by

$2\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}+x_{5} x_{1}\right)$

Write down the matrix $A$ for the corresponding symmetric bilinear form, and calculate $\operatorname{det} A$. Hence, or otherwise, find the rank and signature of $q$.

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• # 2.I.8B

Let $V$ be a finite-dimensional vector space over a field $k$. Describe a bijective correspondence between the set of bilinear forms on $V$, and the set of linear maps of $V$ to its dual space $V^{*}$. If $\phi_{1}, \phi_{2}$ are non-degenerate bilinear forms on $V$, prove that there exists an isomorphism $\alpha: V \rightarrow V$ such that $\phi_{2}(u, v)=\phi_{1}(u, \alpha v)$ for all $u, v \in V$. If furthermore both $\phi_{1}, \phi_{2}$ are symmetric, show that $\alpha$ is self-adjoint (i.e. equals its adjoint) with respect to $\phi_{1}$.

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• # 2.II.17B

Suppose $p$ is an odd prime and $a$ an integer coprime to $p$. Define the Legendre symbol $\left(\frac{a}{p}\right)$, and state (without proof) Euler's criterion for its calculation.

For $j$ any positive integer, we denote by $r_{j}$ the (unique) integer with $\left|r_{j}\right| \leq(p-1) / 2$ and $r_{j} \equiv a j \bmod p$. Let $l$ be the number of integers $1 \leq j \leq(p-1) / 2$ for which $r_{j}$ is negative. Prove that

$\left(\frac{a}{p}\right)=(-1)^{l} .$

Hence determine the odd primes for which 2 is a quadratic residue.

Suppose that $p_{1}, \ldots, p_{m}$ are primes congruent to 7 modulo 8 , and let

$N=8\left(p_{1} \cdots p_{m}\right)^{2}-1$

Show that 2 is a quadratic residue for any prime dividing $N$. Prove that $N$ is divisible by some prime $p \equiv 7 \bmod 8$. Hence deduce that there are infinitely many primes congruent to 7 modulo 8 .

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• # 3.I.9B

Let $A$ be the Hermitian matrix

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

Explaining carefully the method you use, find a diagonal matrix $D$ with rational entries, and an invertible (complex) matrix $T$ such that $T^{*} D T=A$, where $T^{*}$ here denotes the conjugated transpose of $T$.

Explain briefly why we cannot find $T, D$ as above with $T$ unitary.

[You may assume that if a monic polynomial $t^{3}+a_{2} t^{2}+a_{1} t+a_{0}$ with integer coefficients has all its roots rational, then all its roots are in fact integers.]

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• # 3.II.19B

Let $J_{1}$ denote the $2 \times 2$ matrix $\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right)$. Suppose that $T$ is a $2 \times 2$ uppertriangular real matrix with strictly positive diagonal entries and that $J_{1}^{-1} T J_{1} T^{-1}$ is orthogonal. Verify that $J_{1} T=T J_{1}$.

Prove that any real invertible matrix $A$ has a decomposition $A=B C$, where $B$ is an orthogonal matrix and $C$ is an upper-triangular matrix with strictly positive diagonal entries.

Let $A$ now denote a $2 n \times 2 n$ real matrix, and $A=B C$ be the decomposition of the previous paragraph. Let $K$ denote the $2 n \times 2 n$ matrix with $n$ copies of $J_{1}$ on the diagonal, and zeros elsewhere, and suppose that $K A=A K$. Prove that $K^{-1} C K C^{-1}$ is orthogonal. From this, deduce that the entries of $K^{-1} C K C^{-1}$ are zero, apart from $n$ orthogonal $2 \times 2$ blocks $E_{1}, \ldots, E_{n}$ along the diagonal. Show that each $E_{i}$ has the form $J_{1}{ }^{-1} C_{i} J_{1} C_{i}^{-1}$, for some $2 \times 2$ upper-triangular matrix $C_{i}$ with strictly positive diagonal entries. Deduce that $K C=C K$ and $K B=B K$.

[Hint: The invertible $2 n \times 2 n$ matrices $S$ with $2 \times 2$ blocks $S_{1}, \ldots, S_{n}$ along the diagonal, but with all other entries below the diagonal zero, form a group under matrix multiplication.]

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