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]