Section A

• 1. Define $$U,W\leq \R ^{4}$$ by $$U=\Span {(1,2,2,1),(1,3,1,3)}$$, $$W=\Span {(1,2,3,4)}$$.

Is it true that $$U\oplus W=\R ^{4}$$? Justify your answer. [4]

• 2. Let $$V$$ be a vector space, $$U\leq V$$ and $$q:V\to V/U$$ the quotient map.

Under what condition on $$U$$ is $$q$$ an isomorphism? [4]

• 3. Let $$p=x^{17}+5x+1\in \R [x]$$, $$v=(1,0,0)\in \R ^{3}$$ and $$\phi =\phi _A\in L(\R ^3)$$, where

\begin{equation*} A= \begin{pmatrix} 1&5&7\\0&2&9\\0&0&6 \end {pmatrix}. \end{equation*}

Compute $$p(\phi )(v)$$. [4]

• 4. Let $$\phi \in L(V)$$ be a linear operator on a finite-dimensional vector space $$V$$ and suppose that

\begin{equation*} \Delta _{\phi }=(x-1)^{3}(x-17)^{2},\qquad m_{\phi }=(x-1)(x-17)^2. \end{equation*}

What is the Jordan normal form of $$\phi$$? [4]

• 5. What is the dual space $$V^{*}$$ of a vector space $$V$$ over a field $$\F$$?

Define $$\alpha ,\beta \in (\R ^3)^{*}$$ by

\begin{align*} \alpha (x)&=x_1+2x_2-x_3,\\ \beta (x)&=3x_1-3x_2. \end{align*} Write down a basis for $$\operatorname {sol} E$$ where $$E=\Span {\alpha ,\beta }$$. [4]

• 6. For which $$t\in \R$$ does the quadratic form $$q_t:\R ^2\to \R$$ given by

\begin{equation*} q_t(x)=x_{1}^{2}+2tx_1x_2-7x_2^2 \end{equation*}

have signature $$(1,1)$$? [4]

Section B

• 7. Let $$\phi :V\to W$$ be a linear map of vector spaces and $$A\leq W$$. Define $$\phi ^{-1}(A)$$ by

\begin{equation*} \phi ^{-1}(A)=\set {v\in V\st \phi (v)\in A}. \end{equation*}

• (a) Show that $$\ker \phi \leq \phi ^{-1}(A)\leq V$$. [6]

• (b) Let $$U\leq V$$ and $$q:V\to V/U$$ be the quotient map.

• (i) Let $$U\leq B\leq V$$. Show that there is a subspace $$A\leq V/U$$ such that $$B=q^{-1}(A)$$. [6]

• (ii) Let $$A_1,A_2\leq V/U$$ and suppose that $$q^{-1}(A_1)=q^{-1}(A_2)$$.

Prove that $$A_1=A_2$$. [6]

• 8.

• (a) Let $$\phi \in L(\C )$$ be a linear operator on a finite-dimensional complex vector space.

• (i) What is the minimum polynomial of $$\phi$$?

• (ii) Show that the roots of the minimum polynomial are precisely the eigenvalues of $$\phi$$.

(You may assume the Cayley–Hamilton theorem without proof.)

[9]

• (b) Let $$\phi =\phi _{A}\in L(\C ^{3})$$ where

\begin{equation*} A= \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ -2 & -3 & 0 \end {pmatrix}. \end{equation*}

• (i) What is the minimum polynomial of $$A$$?

• (ii) What is the Jordan normal form of $$A$$?

[9]

• 9.

• (a) Which of the following are possible signatures of a quadratic form $$q:\R ^4\to \R$$?

• (i) $$(3,0)$$.

• (ii) $$(4,1)$$.

• (iii) $$(2,-2)$$.

In each case, briefly justify your answer. [6]

• (b) Find an invertible matrix $$P$$ such that $$P^TAP$$ is diagonal where

\begin{equation*} A= \begin{pmatrix} 1&2&1\\2&0&-1\\1&-1&1 \end {pmatrix}. \end{equation*}

[12]