• # 1.I.1C

Show, using the summation convention or otherwise, that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-$ (a.b)c, for a, b, c $\in \mathbb{R}^{3}$

The function $\Pi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ is defined by $\Pi(\mathbf{x})=\mathbf{n} \times(\mathbf{x} \times \mathbf{n})$ where $\mathbf{n}$ is a unit vector in $\mathbb{R}^{3}$. Show that $\Pi$ is linear and find the elements of a matrix $P$ such that $\Pi(\mathbf{x})=P \mathbf{x}$ for all $\mathbf{x} \in \mathbb{R}^{3}$.

Find all solutions to the equation $\Pi(\mathbf{x})=\mathbf{x}$. Evaluate $\Pi(\mathbf{n})$. Describe the function П geometrically. Justify your answer.

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• # 1.I.2C

Define what is meant by the statement that the vectors $\mathbf{x}_{1}, \ldots, \mathbf{x}_{n} \in \mathbb{R}^{m}$ are linearly independent. Determine whether the following vectors $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3} \in \mathbb{R}^{3}$ are linearly independent and justify your answer.

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

For the vectors $\mathbf{x}, \mathbf{y}, \mathbf{z}$ taken from a real vector space $V$ consider the statements A) $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly dependent, B) $\exists \alpha, \beta, \gamma \in \mathbb{R}: \alpha \mathbf{x}+\beta \mathbf{y}+\gamma \mathbf{z}=\mathbf{0}$, C) $\exists \alpha, \beta, \gamma \in \mathbb{R}$, not all $=0: \alpha \mathbf{x}+\beta \mathbf{y}+\gamma \mathbf{z}=\mathbf{0}$, D) $\exists \alpha, \beta \in \mathbb{R}$, not both $=0: \mathbf{z}=\alpha \mathbf{x}+\beta \mathbf{y}$, E) $\exists \alpha, \beta \in \mathbb{R}: \mathbf{z}=\alpha \mathbf{x}+\beta \mathbf{y}$, F) $\nexists$ basis of $V$ that contains all 3 vectors $\mathbf{x}, \mathbf{y}, \mathbf{z}$.

State if the following implications are true or false (no justification is required): i) $\mathrm{A} \Rightarrow \mathrm{B}$, vi) $\mathrm{B} \Rightarrow \mathrm{A}$, ii) $\mathrm{A} \Rightarrow \mathrm{C}$, vii) $\mathrm{C} \Rightarrow \mathrm{A}$, iii) $\mathrm{A} \Rightarrow \mathrm{D}$, viii) $\mathrm{D} \Rightarrow \mathrm{A}$, iv) $\mathrm{A} \Rightarrow \mathrm{E}$, ix) $\mathrm{E} \Rightarrow \mathrm{A}$, v) $\mathrm{A} \Rightarrow \mathrm{F}$, x) $\quad \mathrm{F} \Rightarrow \mathrm{A}$.

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• # 1.II.5C

The matrix

$A_{\alpha}=\left(\begin{array}{ccc} 1 & -1 & 2 \alpha+1 \\ 1 & \alpha-1 & 1 \\ 1+\alpha & -1 & \alpha^{2}+4 \alpha+1 \end{array}\right)$

defines a linear map $\Phi_{\alpha}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by $\Phi_{\alpha}(\mathbf{x})=A_{\alpha} \mathbf{x}$. Find a basis for the kernel of $\Phi_{\alpha}$ for all values of $\alpha \in \mathbb{R}$.

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ and $\mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \mathbf{c}_{3}\right\}$ be bases of $\mathbb{R}^{3}$. Show that there exists a matrix $S$, to be determined in terms of $\mathcal{B}$ and $\mathcal{C}$, such that, for every linear mapping $\Phi$, if $\Phi$ has matrix $A$ with respect to $\mathcal{B}$ and matrix $A^{\prime}$ with respect to $\mathcal{C}$, then $A^{\prime}=S^{-1} A S$.

For the bases

$\mathcal{B}=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)\right\}, \mathcal{C}=\left\{\left(\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right),\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right),\left(\begin{array}{l} 2 \\ 3 \\ 2 \end{array}\right)\right\},$

find the basis transformation matrix $S$ and calculate $S^{-1} A_{0} S$.

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• # 1.II.6C

Assume that $\mathbf{x}_{p}$ is a particular solution to the equation $A \mathbf{x}=\mathbf{b}$ with $\mathbf{x}, \mathbf{b} \in \mathbb{R}^{3}$ and a real $3 \times 3$ matrix $A$. Explain why the general solution to $A \mathbf{x}=\mathbf{b}$ is given by $\mathbf{x}=\mathbf{x}_{p}+\mathbf{h}$ where $\mathbf{h}$ is any vector such that $A \mathbf{h}=\mathbf{0}$.

Now assume that $A$ is a real symmetric $3 \times 3$ matrix with three different eigenvalues $\lambda_{1}, \lambda_{2}$ and $\lambda_{3}$. Show that eigenvectors of $A$ with respect to different eigenvalues are orthogonal. Let $\mathbf{x}_{k}$ be a normalised eigenvector of $A$ with respect to the eigenvalue $\lambda_{k}$, $k=1,2,3$. Show that the linear system

$\left(A-\lambda_{k} I\right) \mathbf{x}=\mathbf{b}$

where $I$ denotes the $3 \times 3$ unit matrix, is solvable if and only if $\mathbf{x}_{k} \cdot \mathbf{b}=0$. Show that the general solution is given by

$\mathbf{x}=\sum_{i \neq k} \frac{\mathbf{b} . \mathbf{x}_{i}}{\lambda_{i}-\lambda_{k}} \mathbf{x}_{i}+\beta \mathbf{x}_{k}, \quad \beta \in \mathbb{R}$

[Hint: consider the components of $\mathbf{x}$ and $\mathbf{b}$ with respect to a basis of eigenvectors of $A$.]

Consider the matrix $A$ and the vector $\mathbf{b}$

$A=\left(\begin{array}{ccc} -\frac{1}{2} \sqrt{2}+\frac{1}{6} \sqrt{3} & \frac{1}{2} \sqrt{2}+\frac{1}{6} \sqrt{3} & -\frac{1}{3} \sqrt{3} \\ \frac{1}{2} \sqrt{2}+\frac{1}{6} \sqrt{3} & -\frac{1}{2} \sqrt{2}+\frac{1}{6} \sqrt{3} & -\frac{1}{3} \sqrt{3} \\ -\frac{1}{3} \sqrt{3} & -\frac{1}{3} \sqrt{3} & \frac{2}{3} \sqrt{3} \end{array}\right), \quad \mathbf{b}=\left(\begin{array}{c} \sqrt{2}+\sqrt{3} \\ -\sqrt{2}+\sqrt{3} \\ -2 \sqrt{3} \end{array}\right)$

Verify that $\frac{1}{\sqrt{3}}(1,1,1)^{T}$ and $\frac{1}{\sqrt{2}}(1,-1,0)^{T}$ are eigenvectors of $A$. Show that $A \mathbf{x}=\mathbf{b}$ is solvable and find its general solution.

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• # 1.II.7C

For $\alpha, \gamma \in \mathbb{R}, \alpha \neq 0, \beta \in \mathbb{C}$ and $\beta \bar{\beta} \geqslant \alpha \gamma$ the equation $\alpha z \bar{z}-\beta \bar{z}-\bar{\beta} z+\gamma=0$ describes a circle $C_{\alpha \beta \gamma}$ in the complex plane. Find its centre and radius. What does the equation describe if $\beta \bar{\beta}<\alpha \gamma$ ? Sketch the circles $C_{\alpha \beta \gamma}$ for $\beta=\gamma=1$ and $\alpha=-2,-1,-\frac{1}{2}, \frac{1}{2}, 1$.

Show that the complex function $f(z)=\beta \bar{z} / \bar{\beta}$ for $\beta \neq 0$ satisfies $f\left(C_{\alpha \beta \gamma}\right)=C_{\alpha \beta \gamma}$.

[Hint: $f(C)=C$ means that $f(z) \in C \forall z \in C$ and $\forall w \in C \quad \exists z \in C$ such that $f(z)=w .]$

For two circles $C_{1}$ and $C_{2}$ a function $m\left(C_{1}, C_{2}\right)$ is defined by

$m\left(C_{1}, C_{2}\right)=\max _{z \in C_{1}, w \in C_{2}}|z-w|$

Prove that $m\left(C_{1}, C_{2}\right) \leqslant m\left(C_{1}, C_{3}\right)+m\left(C_{2}, C_{3}\right)$. Show that

$m\left(C_{\alpha_{1} \beta_{1} \gamma_{1}}, C_{\alpha_{2} \beta_{2} \gamma_{2}}\right)=\frac{\left|\alpha_{1} \beta_{2}-\alpha_{2} \beta_{1}\right|}{\left|\alpha_{1} \alpha_{2}\right|}+\frac{\sqrt{\beta_{1} \overline{\beta_{1}}-\alpha_{1} \gamma_{1}}}{\left|\alpha_{1}\right|}+\frac{\sqrt{\beta_{2} \overline{\beta_{2}}-\alpha_{2} \gamma_{2}}}{\left|\alpha_{2}\right|}$

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• # 1.II.8C

Let $l_{\mathbf{x}}$ denote the straight line through $\mathbf{x}$ with directional vector $\mathbf{u} \neq \mathbf{0}$

$l_{\mathbf{x}}=\left\{\mathbf{y} \in \mathbb{R}^{3}: \mathbf{y}=\mathbf{x}+\lambda \mathbf{u}, \lambda \in \mathbb{R}\right\}$

Show that $l_{\mathbf{0}}$ is a subspace of $\mathbb{R}^{3}$ and show that $l_{\mathbf{x}_{1}}=l_{\mathbf{x}_{2}} \Leftrightarrow \mathbf{x}_{\mathbf{1}}=\mathbf{x}_{2}+\lambda \mathbf{u}$ for some $\lambda \in \mathbb{R}$.

For fixed $\mathbf{u} \neq \mathbf{0}$ let $\mathcal{L}$ be the set of all the parallel straight lines $l_{\mathbf{x}}\left(\mathbf{x} \in \mathbb{R}^{3}\right)$ with directional vector $\mathbf{u}$. On $\mathcal{L}$ an addition and a scalar multiplication are defined by

$l_{\mathbf{x}}+l_{\mathbf{y}}=l_{\mathbf{x}+\mathbf{y}}, \alpha l_{\mathbf{x}}=l_{\alpha \mathbf{x}}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^{3}, \alpha \in \mathbb{R}$

Explain why these operations are well-defined. Show that the addition is associative and that there exists a zero vector which should be identified.

You may now assume that $\mathcal{L}$ is a vector space. If $\left\{\mathbf{u}, \mathbf{b}_{1}, \mathbf{b}_{2}\right\}$ is a basis for $\mathbb{R}^{3}$ show that $\left\{l_{\mathbf{b}_{1}}, l_{\mathbf{b}_{2}}\right\}$ is a basis for $\mathcal{L}$.

For $\mathbf{u}=(1,3,-1)^{T}$ a linear map $\Phi: \mathcal{L} \rightarrow \mathcal{L}$ is defined by

$\Phi\left(l_{(1,-1,0)^{T}}\right)=l_{(2,4,-1)^{T}}, \Phi\left(l_{(1,1,0)^{T}}\right)=l_{(-4,-2,1)^{T}}$

Find the matrix $A$ of $\Phi$ with respect to the basis $\left\{l_{(1,0,0)^{T}}, l_{(0,1,0)^{T}}\right\}$.

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• # 1.I $. 3 \mathrm{D} \quad$

What does it mean to say that $u_{n} \rightarrow l$ as $n \rightarrow \infty$ ?

Show that, if $u_{n} \rightarrow l$ and $v_{n} \rightarrow k$, then $u_{n} v_{n} \rightarrow l k$ as $n \rightarrow \infty$.

If further $u_{n} \neq 0$ for all $n$ and $l \neq 0$, show that $1 / u_{n} \rightarrow 1 / l$ as $n \rightarrow \infty$.

Give an example to show that the non-vanishing of $u_{n}$ for all $n$ need not imply the non-vanishing of $l$.

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• # 1.I.4D

Starting from the theorem that any continuous function on a closed and bounded interval attains a maximum value, prove Rolle's Theorem. Deduce the Mean Value Theorem.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. If $f^{\prime}(t)>0$ for all $t$ show that $f$ is a strictly increasing function.

Conversely, if $f$ is strictly increasing, is $f^{\prime}(t)>0$ for all $t$ ?

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• # 1.II.10D

Suppose that $f$ is a continuous real-valued function on $[a, b]$ with $f(a). If $f(a) show that there exists $c$ with $a and $f(c)=v$.

Deduce that if $f$ is a continuous function from the closed bounded interval $[a, b]$ to itself, there exists at least one fixed point, i.e., a number $d$ belonging to $[a, b]$ with $f(d)=d$. Does this fixed point property remain true if $f$ is a continuous function defined (i) on the open interval $(a, b)$ and (ii) on $\mathbb{R}$ ? Justify your answers.

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• # 1.II.11D

(i) Show that if $g: \mathbb{R} \rightarrow \mathbb{R}$ is twice continuously differentiable then, given $\epsilon>0$, we can find some constant $L$ and $\delta(\epsilon)>0$ such that

$\left|g(t)-g(\alpha)-g^{\prime}(\alpha)(t-\alpha)\right| \leq L|t-\alpha|^{2}$

for all $|t-\alpha|<\delta(\epsilon)$.

(ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be twice continuously differentiable on $[a, b]$ (with one-sided derivatives at the end points), let $f^{\prime}$ and $f^{\prime \prime}$ be strictly positive functions and let $f(a)<0.

If $F(t)=t-\left(f(t) / f^{\prime}(t)\right)$ and a sequence $\left\{x_{n}\right\}$ is defined by $b=x_{0}, x_{n}=$ $F\left(x_{n-1}\right) \quad(n>0)$, show that $x_{0}, x_{1}, x_{2}, \ldots$ is a decreasing sequence of points in $[a, b]$ and hence has limit $\alpha$. What is $f(\alpha)$ ? Using part (i) or otherwise estimate the rate of convergence of $x_{n}$ to $\alpha$, i.e., the behaviour of the absolute value of $\left(x_{n}-\alpha\right)$ for large values of $n$.

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• # 1.II.12D

Explain what it means for a function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable on $[a, b]$, and give an example of a bounded function that is not Riemann integrable.

Show each of the following statements is true for continuous functions $f$, but false for general Riemann integrable functions $f$.

(i) If $f:[a, b] \rightarrow \mathbb{R}$ is such that $f(t) \geq 0$ for all $t$ in $[a, b]$ and $\int_{a}^{b} f(t) d t=0$, then $f(t)=0$ for all $t$ in $[a, b]$.

(ii) $\int_{a}^{t} f(x) d x$ is differentiable and $\frac{d}{d t} \int_{a}^{t} f(x) d x=f(t)$.

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• # 1.II.9D

(i) If $a_{0}, a_{1}, \ldots$ are complex numbers show that if, for some $w \in \mathbb{C}, w \neq 0$, the set $\left\{\left|a_{n} w^{n}\right|: n \geq 0\right\}$ is bounded and $|z|<|w|$, then $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges absolutely. Use this result to define the radius of convergence of the power series $\sum_{n=0}^{\infty} a_{n} z^{n}$.

(ii) If $\left|a_{n}\right|^{1 / n} \rightarrow R$ as $n \rightarrow \infty(0 show that $\sum_{n=0}^{\infty} a_{n} z^{n}$ has radius of convergence equal to $1 / R$.

(iii) Give examples of power series with radii of convergence 1 such that (a) the series converges at all points of the circle of convergence, (b) diverges at all points of the circle of convergence, and (c) neither of these occurs.

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