Part IA, 2002, Paper 1

Algebra and Geometry

• 1.I.1B

  Algebra and Geometry | Part IA, 2002

  (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 $T$ ?

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• 1.I.2D

  Algebra and Geometry | Part IA, 2002

  State 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.

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• 1.II.5B

  Algebra and Geometry | Part IA, 2002

  (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 $H \neq G$.

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• 1.II.6B

  Algebra and Geometry | Part IA, 2002

  (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.

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• 1.II.7B

  Algebra and Geometry | Part IA, 2002

  (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.

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• 1.II.8D

  Algebra and Geometry | Part IA, 2002

  Define 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,

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

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Analysis I

• 1.I $. 3 C$

  Analysis I | Part IA, 2002

  Suppose $a_{n} \in \mathbb{R}$ 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 $n \rightarrow \infty$.

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• 1.I.4C

  Analysis I | Part IA, 2002

  Show 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.

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• 1.II.10C

  Analysis I | Part IA, 2002

  Show 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.]

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• 1.II.11C

  Analysis I | Part IA, 2002

  State 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.

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• 1.II.12C

  Analysis I | Part IA, 2002

  Prove 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

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

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• 1.II.9C

  Analysis I | Part IA, 2002

  State 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 $0 \leqslant r \leqslant 2^{n}-1, n \geqslant 0$.

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