1. Let \(V\) be a vector space over a field \(\F \) and \(U,W\leq V\) vector subspaces.

What does it mean to say that \(V\) is the internal direct sum of \(U\) and \(W\), that is, \(V=U\oplus W\)?

What is a complement to \(U\) in \(V\)? [4]

2. Let \(V\) be an \(n\)-dimensional vector space over a field \(\F \) and \(U,W\leq V\) with \(\dim U=1\) and \(\dim W=n-1\).

Show that either \(U\leq W\) or \(V=U\oplus W\). [4]

3. Let \(V\) be a finite-dimensional inner product space.

What is an orthonormal basis of \(V\)?

Let \(\lst {u}1n\) be an orthonormal basis of \(V\) and \(v,w\in V\). Prove that

\begin{equation*} v=\sum _{i=1}^n\ip {u_i,v}u_i \end{equation*}

and so deduce that

\begin{equation*} \ip {v,w}=\sum _{i=1}^n\ip {v,u_i}\ip {u_i,w}. \end{equation*}

[4]

4. Let \(V\) be a finite-dimensional complex inner product space and \(\phi :V\to V\) a linear operator.

What is an adjoint of \(\phi \)?

What does it mean to say that \(\phi \) is normal?

Show that \(\phi \) is normal if and only if, for all \(v,w\in V\),

\begin{equation*} \ip {\phi (v),\phi (w)}=\ip {\phi ^{*}(v),\phi ^{*}(w)}. \end{equation*}

[4]

5. Let \(V\) be a vector space over a field \(\F \).

What is the dual space \(V^{*}\)?

Let \(\lst {v}1n\) be a basis of \(V\). Define the dual basis \(\lst {v^{*}}1n\) of \(V^{*}\) to \(\lst {v}1n\).

Prove that, for \(\alpha \in V^{*}\),

\begin{equation*} \alpha =\sum _{i=1}^n\alpha (v_i)v^{*}_i. \end{equation*}

[4]

6. What is the rank and signature of the quadratic form \(Q:\R ^2\to \R \) given by

\begin{equation*} Q(x)=x_1^2+x_1x_2+x_2^2. \end{equation*}

[4]

7.

• (a) State the First Isomorphism Theorem. [4]

• (b) Let \(\phi :V\to W\) be a linear surjection of vector spaces over a field \(\F \) and \(U=\ker \phi \leq V\).

  Prove that \(V/U\cong W\). [4]

• (c) Let \(V\) be a vector space over a field \(\F \) and \(U\leq V\) a subspace such that \(V/U\) is finite-dimensional.

  Show that \(U\) has a complement \(W\) in \(V\) with \(\dim W=\dim V/U\).

  Hint: For \(q:V\to V/U\) the quotient map, let \(q(v_1),\dots ,q(v_n)\) be a basis of \(V/U\) and consider the span of \(\lst {v}1n\). [10]

8.

• (a) State and prove the Riesz Representation Theorem. [6]

• (b) Let \(U\) be the subspace of \(\R ^4\) spanned by \((1,-1,0,0)\), \((0,1,-1,0)\) and \((0,0,1,-1)\).

  • (i) Find an orthonormal basis of \(U\). [6]

  • (ii) Stating any results from lectures that you use, find \(u\in U\) such that \(\norm {u-(1,1,2,2)}\) is as small as possible. [6]

9. Let \(A\) be the matrix

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

• (a) Find a matrix \(P\) such that \(P^TAP\) is diagonal. [9]

• (b) Let \(V\) be a real vector space.

  What is a symmetric bilinear form on \(V\)?

  What are the radical, rank and signature of a symmetric bilinear form on \(V\)? [5]

• (c) Compute the rank and signature of the symmetric bilinear form \(B\) on \(\R ^3\) given by

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

  \begin{equation*} B(x,y)=\bx ^TA\mathbf {y}. \end{equation*}

  [4]