Part IA, 2021, Paper 1

Analysis I

• Paper 1, Section I, F

  Analysis I | Part IA, 2021

  State and prove the Bolzano-Weierstrass theorem.

  Consider a bounded sequence $\left(x_{n}\right)$. Prove that if every convergent subsequence of $\left(x_{n}\right)$ converges to the same limit $L$ then $\left(x_{n}\right)$ converges to $L$.

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• Paper 1, Section I, F

  Analysis I | Part IA, 2021

  State and prove the alternating series test. Hence show that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges. Show also that

  $$\frac{7}{12} \leqslant \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \leqslant \frac{47}{60}$$

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• Paper 1, Section II, F

  Analysis I | Part IA, 2021

  (a) Let $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with $a_{n} \in \mathbb{C}$. Show that there exists $R \in[0, \infty]$ (called the radius of convergence) such that the series is absolutely convergent when $|z|<R$ but is divergent when $|z|>R$.

  Suppose that the radius of convergence of the series $\sum_{n=0}^{\infty} a_{n} z^{n}$ is $R=2$. For a fixed positive integer $k$, find the radii of convergence of the following series. [You may assume that $\lim _{n \rightarrow \infty}\left|a_{n}\right|^{1 / n}$ exists.] (i) $\sum_{n=0}^{\infty} a_{n}^{k} z^{n}$. (ii) $\sum_{n=0}^{\infty} a_{n} z^{k n}$. (iii) $\sum_{n=0}^{\infty} a_{n} z^{n^{2}}$.

  (b) Suppose that there exist values of $z$ for which $\sum_{n=0}^{\infty} b_{n} e^{n z}$ converges and values for which it diverges. Show that there exists a real number $S$ such that $\sum_{n=0}^{\infty} b_{n} e^{n z}$ diverges whenever $\operatorname{Re}(z)>S$ and converges whenever $\operatorname{Re}(z)<S$.

  Determine the set of values of $z$ for which

  $$\sum_{n=0}^{\infty} \frac{2^{n} e^{i n z}}{(n+1)^{2}}$$

  converges.

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• Paper 1, Section II, F

  Analysis I | Part IA, 2021

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be $n$-times differentiable, for some $n>0$.

  (a) State and prove Taylor's theorem for $f$, with the Lagrange form of the remainder. [You may assume Rolle's theorem.]

  (b) Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is an infinitely differentiable function such that $f(0)=1$ and $f^{\prime}(0)=0$, and satisfying the differential equation $f^{\prime \prime}(x)=-f(x)$. Prove carefully that

  $$f(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !}$$

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• Paper 1, Section II, F

  Analysis I | Part IA, 2021

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function.

  (a) Let $m=\min _{x \in[a, b]} f(x)$ and $M=\max _{x \in[a, b]} f(x)$. If $g:[a, b] \rightarrow \mathbb{R}$ is a positive continuous function, prove that

  $$m \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x$$

  directly from the definition of the Riemann integral.

  (b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Show that

  $$\int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x \rightarrow f(0)$$

  as $n \rightarrow \infty$, and deduce that

  $$\int_{0}^{1} n f(x) e^{-n x} d x \rightarrow f(0)$$

  as $n \rightarrow \infty$

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• Paper 1, Section II, F

  Analysis I | Part IA, 2021

  (a) State the intermediate value theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous bijection and $x_{1}<x_{2}<x_{3}$ then either $f\left(x_{1}\right)<f\left(x_{2}\right)<f\left(x_{3}\right)$ or $f\left(x_{1}\right)>f\left(x_{2}\right)>f\left(x_{3}\right)$. Deduce that $f$ is either strictly increasing or strictly decreasing.

  (b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions. Which of the following statements are true, and which can be false? Give a proof or counterexample as appropriate.

  (i) If $f$ and $g$ are continuous then $f \circ g$ is continuous.

  (ii) If $g$ is strictly increasing and $f \circ g$ is continuous then $f$ is continuous.

  (iii) If $f$ is continuous and a bijection then $f^{-1}$ is continuous.

  (iv) If $f$ is differentiable and a bijection then $f^{-1}$ is differentiable.

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Vectors and Matrices

• Paper 1, Section I, B

  Vectors and Matrices | Part IA, 2021

  The matrix

  $$A=\left(\begin{array}{rr} 2 & -1 \\ 2 & 0 \\ -1 & 1 \end{array}\right)$$

  represents a linear map $\Phi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ with respect to the bases

  $$B=\left\{\left(\begin{array}{l} 0 \\ 2 \end{array}\right),\left(\begin{array}{r} -2 \\ 0 \end{array}\right)\right\}, \quad C=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)\right\}$$

  Find the matrix $A^{\prime}$ that represents $\Phi$ with respect to the bases

  $$B^{\prime}=\left\{\left(\begin{array}{l} 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -1 \end{array}\right)\right\}, \quad C^{\prime}=\left\{\left(\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right)\right\}$$

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• Paper 1, Section I, C

  Vectors and Matrices | Part IA, 2021

  (a) Find all complex solutions to the equation $z^{i}=1$.

  (b) Write down an equation for the numbers $z$ which describe, in the complex plane, a circle with radius 5 centred at $c=5 i$. Find the points on the circle at which it intersects the line passing through $c$ and $z_{0}=\frac{15}{4}$.

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• Paper 1, Section II, 8B

  Vectors and Matrices | Part IA, 2021

  (a) Consider the matrix

  $$A=\left(\begin{array}{rrr} \mu & 1 & 1 \\ 2 & -\mu & 0 \\ -\mu & 2 & 1 \end{array}\right)$$

  Find the kernel of $A$ for each real value of the constant $\mu$. Hence find how many solutions $\mathbf{x} \in \mathbb{R}^{3}$ there are to

  $$A \mathbf{x}=\left(\begin{array}{l} 1 \\ 1 \\ 2 \end{array}\right)$$

  depending on the value of $\mu$. [There is no need to find expressions for the solution(s).]

  (b) Consider the reflection map $\Phi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ defined as

  $$\Phi: \mathbf{x} \mapsto \mathbf{x}-2(\mathbf{x} \cdot \mathbf{n}) \mathbf{n}$$

  where $\mathbf{n}$ is a unit vector normal to the plane of reflection.

  (i) Find the matrix $H$ which corresponds to the map $\Phi$ in terms of the components of $\mathbf{n}$.

  (ii) Prove that a reflection in a plane with unit normal $\mathbf{n}$ followed by a reflection in a plane with unit normal vector $\mathbf{m}$ (both containing the origin) is equivalent to a rotation along the line of intersection of the planes with an angle twice that between the planes.

  [Hint: Choose your coordinate axes carefully.]

  (iii) Briefly explain why a rotation followed by a reflection or vice-versa can never be equivalent to another rotation.

  Part IA, 2021 List of Questions

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2021

  Let $A$ be a real, symmetric $n \times n$ matrix.

  We say that $A$ is positive semi-definite if $\mathbf{x}^{T} A \mathbf{x} \geqslant 0$ for all $\mathbf{x} \in \mathbb{R}^{n}$. Prove that $A$ is positive semi-definite if and only if all the eigenvalues of $A$ are non-negative. [You may quote results from the course, provided that they are clearly stated.]

  We say that $A$ has a principal square root $B$ if $A=B^{2}$ for some symmetric, positive semi-definite $n \times n$ matrix $B$. If such a $B$ exists we write $B=\sqrt{A}$. Show that if $A$ is positive semi-definite then $\sqrt{A}$ exists.

  Let $M$ be a real, non-singular $n \times n$ matrix. Show that $M^{T} M$ is symmetric and positive semi-definite. Deduce that $\sqrt{M^{T} M}$ exists and is non-singular. By considering the matrix

  $$M\left(\sqrt{M^{T} M}\right)^{-1}$$

  or otherwise, show $M=R P$ for some orthogonal $n \times n$ matrix $R$ and a symmetric, positive semi-definite $n \times n$ matrix $P$.

  Describe the transformation $R P$ geometrically in the case $n=3$.

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• Paper 1, Section II, A

  Vectors and Matrices | Part IA, 2021

  (a) For an $n \times n$ matrix $A$ define the characteristic polynomial $\chi_{A}$ and the characteristic equation.

  The Cayley-Hamilton theorem states that every $n \times n$ matrix satisfies its own characteristic equation. Verify this in the case $n=2$.

  (b) Define the adjugate matrix $\operatorname{adj}(A)$ of an $n \times n$ matrix $A$ in terms of the minors of $A$. You may assume that

  $$A \operatorname{adj}(A)=\operatorname{adj}(A) A=\operatorname{det}(A) I$$

  where $I$ is the $n \times n$ identity matrix. Show that if $A$ and $B$ are non-singular $n \times n$ matrices then

  $$\operatorname{adj}(A B)=\operatorname{adj}(B) \operatorname{adj}(A)$$

  (c) Let $M$ be an arbitrary $n \times n$ matrix. Explain why

  (i) there is an $\alpha>0$ such that $M-t I$ is non-singular for $0<t<\alpha$;

  (ii) the entries of $\operatorname{adj}(M-t I)$ are polynomials in $t$.

  Using parts (i) and (ii), or otherwise, show that $(*)$ holds for all matrices $A, B$.

  (d) The characteristic polynomial of the arbitrary $n \times n$ matrix $A$ is

  $$\chi_{A}(z)=(-1)^{n} z^{n}+c_{n-1} z^{n-1}+\cdots+c_{1} z+c_{0}$$

  By considering adj $(A-t I)$, or otherwise, show that

  $$\operatorname{adj}(A)=(-1)^{n-1} A^{n-1}-c_{n-1} A^{n-2}-\cdots-c_{2} A-c_{1} I .$$

  [You may assume the Cayley-Hamilton theorem.]

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• Paper 1, Section II, C

  Vectors and Matrices | Part IA, 2021

  Using the standard formula relating products of the Levi-Civita symbol $\epsilon_{i j k}$ to products of the Kronecker $\delta_{i j}$, prove

  $$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$$

  Define the scalar triple product $[\mathbf{a}, \mathbf{b}, \mathbf{c}]$ of three vectors $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ in $\mathbb{R}^{3}$ in terms of the dot and cross product. Show that

  $$[\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}]=[\mathbf{a}, \mathbf{b}, \mathbf{c}]^{2}$$

  Given a basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ for $\mathbb{R}^{3}$ which is not necessarily orthonormal, let

  $$\mathbf{e}_{1}^{\prime}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \mathbf{e}_{2}^{\prime}=\frac{\mathbf{e}_{3} \times \mathbf{e}_{1}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \mathbf{e}_{3}^{\prime}=\frac{\mathbf{e}_{1} \times \mathbf{e}_{2}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}$$

  Show that $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$ is also a basis for $\mathbb{R}^{3}$. [You may assume that three linearly independent vectors in $\mathbb{R}^{3}$ form a basis.]

  The vectors $\mathbf{e}_{1}^{\prime \prime}, \mathbf{e}_{2}^{\prime \prime}, \mathbf{e}_{3}^{\prime \prime}$ are constructed from $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$ in the same way that $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}$, $\mathbf{e}_{3}^{\prime}$ are constructed from $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Show that

  $$\mathbf{e}_{1}^{\prime \prime}=\mathbf{e}_{1}, \quad \mathbf{e}_{2}^{\prime \prime}=\mathbf{e}_{2}, \quad \mathbf{e}_{3}^{\prime \prime}=\mathbf{e}_{3}$$

  An infinite lattice consists of all points with position vectors given by

  $$\mathbf{R}=n_{1} \mathbf{e}_{1}+n_{2} \mathbf{e}_{2}+n_{3} \mathbf{e}_{3} \text { with } n_{1}, n_{2}, n_{3} \in \mathbb{Z}$$

  Find all points with position vectors $\mathbf{K}$ such that $\mathbf{K} \cdot \mathbf{R}$ is an integer for all integers $n_{1}$, $n_{2}, n_{3}$.

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