Section A

• 1. Let $$V$$ be a vector space over a field $$\F$$ and let $$\lst {V}1k\leq V$$ be linear subspaces.

What is the sum $$\plst {V}1k$$?

What does it mean to say that the sum $$\plst {V}1k$$ is direct? [4]

• 2. Let $$V$$ be a vector space over a field $$\F$$, $$v\in V$$ and $$U\leq V$$ a linear subspace.

What is the coset $$v+U$$?

Let $$w\in V$$. Show that $$w\in v+U$$ if and only if $$\lambda v+(1-\lambda )w\in v+U$$, for all $$\lambda \in \F$$. [4]

• 3. Let $$V$$ be a vector space over $$\C$$.

What is an inner product on $$V$$?

For $$x,y\in \C ^2$$, let

\begin{equation*} \ip {x,y}=\overline {x_{1}}y_2+\overline {x_2}y_1. \end{equation*}

Is $$\ip {\,,\,}$$ an inner product on $$\C ^2$$? (You must justify your answer.) [4]

• 4. Let $$V$$ be an inner product space over $$\C$$ and $$\phi$$ a linear operator on $$V$$.

What is an adjoint of $$\phi$$?

What does it mean to say that $$\phi$$ is normal? [4]

• 5. Let $$V$$ be a vector space over a field $$\F$$.

What is the dual space $$V^{*}$$ of $$V$$?

Let $$U\leq V$$ be a linear subspace. What is the annihilator $$\ann U$$ of $$U$$? [4]

• 6. Compute the rank and signature of the quadratic form $$Q$$ on $$\R ^2$$ given by

\begin{equation*} Q(x)=x_1^2-6x_1x_2+9x_2^2? \end{equation*}

[4]

Section B

• 7. Let $$V,W$$ be finite-dimensional vector spaces over a field $$\F$$, $$U\leq V$$ a linear subspace and $$\phi :U\to W$$ be a linear map.

• (a) Prove that there is a linear map $$\Phi :V\to W$$ such that

\begin{equation*} \Phi (u)=\phi (u), \end{equation*}

for all $$u\in U$$. [6]

• (b) Prove that the restriction map $$r:L(V,W)\to L(U,W)$$ given by $$r(\Phi )=\Phi _{|U}$$ is a linear surjection.

What is its kernel? [6]

• (c) State the First Isomorphism Theorem. [2]

• (d) Prove that $$U^{*}\cong V^{*}/\ann U$$. [4]

• 8.

• (a) State and prove the Cauchy–Schwarz inequality. [6]

• (b) Let $$\lst {a}1n\in \R$$. Prove that

\begin{equation*} \bigl (\sum _{i=1}^na_i/n\bigr )^2\leq \sum _{i=1}^na_i^2/n. \end{equation*}

[3]

• (c) Compute the QR decomposition of the matrix $$A$$ given by

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

[9]

• 9.

• (a) Let $$V$$ be a real vector space.

What is a bilinear form on $$V$$?

What is a quadratic form on $$V$$?

What are the rank and signature of a quadratic form?

State Sylvester’s Law of Inertia. [9]

• (b) Diagonalise the quadratic form $$Q:\R ^3\to \R$$ given by

\begin{equation*} Q(x)=x_1^2+3x^2_2-x_3^2+2x_1x_2+4x_2x_3. \end{equation*}

What are the rank and signature of $$Q$$? [9]