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*}


  • 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*}


    • (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]