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]

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

  \(\seteqnumber{0}{}{0}\)

  \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

  \(\seteqnumber{0}{}{0}\)

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

  [12]