Section A

• 1. Define subspaces $$U,W\leq \R ^3$$ by:

\begin{equation*} U=\Span {(1,1,1),(1,2,0)},\qquad W=\Span {(1,0,2)}. \end{equation*}

Is the sum $$U+W$$ direct? (You must justify your answer.) [4]

• 2. With $$U\leq \R ^{3}$$ as in question 1, $$v=(1,2,3)$$ and $$w=(0,2,1)$$, is $$v\equiv w\mod U$$? (You must justify your answer.) [4]

• 3. For $$x,y\in \C ^{3}$$, let

\begin{equation*} \ip {x,y}=x_1y_1+x_2y_2+\bar {x}_3y_3. \end{equation*}

Is $$\ip {\,\, ,\,}$$ an inner product on $$\C ^3$$? (You must justify your answer.) [4]

• 4. Show that $$A\in M_{2\times 2}(\C )$$ given by

\begin{equation*} \frac {1}{\sqrt {5}} \begin{pmatrix} 1&2i\\2i&1 \end {pmatrix} \end{equation*}

is a unitary matrix and compute its eigenvalues. [4]

• 5. With $$U\leq \R ^3$$ as in question 1, write down a non-zero element of $$\ann U$$.

Hint: any $$\alpha \in (\R ^{3})^{*}$$ is of the form $$\alpha (x)=\alpha _1x_1+\alpha _2x_2+\alpha _3x_3$$. [4]

• 6. Define a quadratic form Q on $$\R ^2$$ by

\begin{equation*} Q(x)=4x_1^2+4x_1x_2+x_2^2. \end{equation*}

Compute the rank and signature of $$Q$$. [4]

Section B

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

• (a) Let $$\lst {V}1k\leq V$$ be subspaces of $$V$$.

Show that the sum $$\plst {V}1k$$ is direct if and only if, whenever $$v_i\in V_i$$, $$1\leq i\leq k$$, satisfy

\begin{equation*} \plst {v}1k=0 \end{equation*}

then each $$v_i=0$$. [6]

• (b) Let $$u,w\in V$$ and $$V_1,V_2\leq V$$ subspaces such that $$u+V_1=w+V_2$$.

Prove that $$V_1=V_2$$. [6]

• (c) Let $$\alpha \in V^{*}$$ be non-zero. Prove that

\begin{equation*} \dim (V/\ker \alpha ) =1. \end{equation*}

[6]

• 8.

• (a) Let $$a,b,c,d,e$$ be positive real numbers. Show that

\begin{equation*} 25\leq (a+b+c+d+e)(1/a+1/b+1/c+1/d+1/e). \end{equation*}

For what $$a,b,c,d,e$$ do we get equality? [5]

• (b) Find the QR decomposition for the matrix $$A$$ given by

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

[7]

• (c) Let $$U=\Span {(1,-1,0),(1,0,-1)}\leq \R ^3$$ and $$v=(2,1,2)$$.

Find the closest point of $$U$$ to $$v$$ (where we define distance in $$\R ^3$$ using the dot product). [6]

• 9. Let $$B$$ be the symmetric bilinear form on $$\R ^3$$ defined by the matrix

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

Thus $$B(x,y)=\mathbf {x}^TA\mathbf {y}$$.

• (a) Find a diagonalising basis for $$B$$. [10]

• (b) Compute the rank and signature of $$B$$. [4]

• (c) Compute the radical of $$B$$. [4]