Part IA, 2002, Paper 1

# Part IA, 2002, Paper 1

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1.I.1B

comment(a) State the Orbit-Stabilizer Theorem for a finite group $G$ acting on a set $X$.

(b) Suppose that $G$ is the group of rotational symmetries of a cube $C$. Two regular tetrahedra $T$ and $T^{\prime}$ are inscribed in $C$, each using half the vertices of $C$. What is the order of the stabilizer in $G$ of $T$ ?

1.I.2D

commentState the Fundamental Theorem of Algebra. Define the characteristic equation for an arbitrary $3 \times 3$ matrix $A$ whose entries are complex numbers. Explain why the matrix must have three eigenvalues, not necessarily distinct.

Find the characteristic equation of the matrix

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

and hence find the three eigenvalues of $A$. Find a set of linearly independent eigenvectors, specifying which eigenvector belongs to which eigenvalue.

1.II.5B

comment(a) Find a subset $T$ of the Euclidean plane $\mathbb{R}^{2}$ that is not fixed by any isometry (rigid motion) except the identity.

Let $G$ be a subgroup of the group of isometries of $\mathbb{R}^{2}, T$ a subset of $\mathbb{R}^{2}$ not fixed by any isometry except the identity, and let $S$ denote the union $\bigcup_{g \in G} g(T)$. Does the group $H$ of isometries of $S$ contain $G$ ? Justify your answer.

(b) Find an example of such a $G$ and $T$ with $H \neq G$.

1.II.6B

comment(a) Suppose that $g$ is a Möbius transformation, acting on the extended complex plane. What are the possible numbers of fixed points that $g$ can have? Justify your answer.

(b) Show that the operation $c$ of complex conjugation, defined by $c(z)=\bar{z}$, is not a Möbius transformation.

1.II.7B

comment(a) Find, with justification, the matrix, with respect to the standard basis of $\mathbb{R}^{2}$, of the rotation through an angle $\alpha$ about the origin.

(b) Find the matrix, with respect to the standard basis of $\mathbb{R}^{3}$, of the rotation through an angle $\alpha$ about the axis containing the point $\left(\frac{3}{5}, \frac{4}{5}, 0\right)$ and the origin. You may express your answer in the form of a product of matrices.

1.II.8D

commentDefine what is meant by a vector space $V$ over the real numbers $\mathbb{R}$. Define subspace, proper subspace, spanning set, basis, and dimension.

Define the sum $U+W$ and intersection $U \cap W$ of two subspaces $U$ and $W$ of a vector space $V$. Why is the intersection never empty?

Let $V=\mathbb{R}^{4}$ and let $U=\left\{\mathbf{x} \in V: x_{1}-x_{2}+x_{3}-x_{4}=0\right\}$, where $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$, and let $W=\left\{\mathbf{x} \in V: x_{1}-x_{2}-x_{3}+x_{4}=0\right\}$. Show that $U \cap W$ has the orthogonal basis $\mathbf{b}_{1}, \mathbf{b}_{2}$ where $\mathbf{b}_{1}=(1,1,0,0)$ and $\mathbf{b}_{2}=(0,0,1,1)$. Extend this basis to find orthogonal bases of $U, W$, and $U+W$. Show that $U+W=V$ and hence verify that, in this case,

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

1.I $. 3 C$

commentSuppose $a_{n} \in \mathbb{R}$ for $n \geqslant 1$ and $a \in \mathbb{R}$. What does it mean to say that $a_{n} \rightarrow a$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$ ?

Show that, if $a_{n} \neq 0$ for all $n$ and $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$, then $1 / a_{n} \rightarrow 0$ as $n \rightarrow \infty$. Is the converse true? Give a proof or a counter example.

Show that, if $a_{n} \neq 0$ for all $n$ and $a_{n} \rightarrow a$ with $a \neq 0$, then $1 / a_{n} \rightarrow 1 / a$ as $n \rightarrow \infty$.

1.I.4C

commentShow that any bounded sequence of real numbers has a convergent subsequence.

Give an example of a sequence of real numbers with no convergent subsequence.

Give an example of an unbounded sequence of real numbers with a convergent subsequence.

1.II.10C

commentShow that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds.

Write down examples of the following functions (no proof is required).

(i) A continuous function $f_{1}:(0,1) \rightarrow \mathbb{R}$ which is not bounded.

(ii) A continuous function $f_{2}:(0,1) \rightarrow \mathbb{R}$ which is bounded but does not attain its bounds.

(iii) A bounded function $f_{3}:[0,1] \rightarrow \mathbb{R}$ which is not continuous.

(iv) A function $f_{4}:[0,1] \rightarrow \mathbb{R}$ which is not bounded on any interval $[a, b]$ with $0 \leqslant a<b \leqslant 1 .$

[Hint: Consider first how to define $f_{4}$ on the rationals.]

1.II.11C

commentState the mean value theorem and deduce it from Rolle's theorem.

Use the mean value theorem to show that, if $h: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable with $h^{\prime}(x)=0$ for all $x$, then $h$ is constant.

By considering the derivative of the function $g$ given by $g(x)=e^{-a x} f(x)$, find all the solutions of the differential equation $f^{\prime}(x)=a f(x)$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and $a$ is a fixed real number.

Show that, if $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then the function $F: \mathbb{R} \rightarrow \mathbb{R}$ given by

$F(x)=\int_{0}^{x} f(t) d t$

is differentiable with $F^{\prime}(x)=f(x)$.

Find the solution of the equation

$g(x)=A+\int_{0}^{x} g(t) d t$

where $g: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and $A$ is a real number. You should explain why the solution is unique.

1.II.12C

commentProve Taylor's theorem with some form of remainder.

An infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the differential equation

$f^{(3)}(x)=f(x)$

and the conditions $f(0)=1, f^{\prime}(0)=f^{\prime \prime}(0)=0$. If $R>0$ and $j$ is a positive integer, explain why we can find an $M_{j}$ such that

$\left|f^{(j)}(x)\right| \leqslant M_{j}$

for all $x$ with $|x| \leqslant R$. Explain why we can find an $M$ such that

$\left|f^{(j)}(x)\right| \leqslant M$

for all $x$ with $|x| \leqslant R$ and all $j \geqslant 0$.

Use your form of Taylor's theorem to show that

$f(x)=\sum_{n=0}^{\infty} \frac{x^{3 n}}{(3 n) !}$

1.II.9C

commentState some version of the fundamental axiom of analysis. State the alternating series test and prove it from the fundamental axiom.

In each of the following cases state whether $\sum_{n=1}^{\infty} a_{n}$ converges or diverges and prove your result. You may use any test for convergence provided you state it correctly.

(i) $a_{n}=(-1)^{n}(\log (n+1))^{-1}$.

(ii) $a_{2 n}=(2 n)^{-2}, a_{2 n-1}=-n^{-2}$.

(iii) $a_{3 n-2}=-(2 n-1)^{-1}, a_{3 n-1}=(4 n-1)^{-1}, a_{3 n}=(4 n)^{-1}$.

(iv) $a_{2^{n}+r}=(-1)^{n}\left(2^{n}+r\right)^{-1}$ for $0 \leqslant r \leqslant 2^{n}-1, n \geqslant 0$.