Section A

• 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]

Section B

• 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

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

[4]