Section A

  • 1. Define subspaces \(U,W\leq \R ^{3}\) by \(U=\Span {(1,0,-1),(2,-1,0)}\) and \(W=\Span {(1,1,1)}\).

    Show that the sum \(U+W\) is direct. [4]

  • 2. With \(U,W\) as in question 1, find \(u\in U\) and \(w\in W\) such that

    \begin{equation*} (1,2,3)=u+w. \end{equation*}


  • 3. Use dot product to make \(\R ^3\) into an inner product space.

    With \(W\) as in question 1, find a basis for \(W^{\perp }\). [4]

  • 4. Let \(V\) be a complex inner product space and \(\phi \) a unitary operator on \(V\).

    If \(\lambda \) is an eigenvalue of \(\phi \), show that \(\abs {\lambda }=1\). [4]

  • 5. Let \(V\) be a vector space over a field \(\F \) and \(v,w\in V\) with \(v\neq w\). Show that there is \(\alpha \in V^{*}\) such that \(\alpha (v)\neq \alpha (w)\). [4]

  • 6. For \(t\in \R \), define a quadratic form \(Q_{t}\) on \(\R ^2\) by \(Q_{t}(x)=x_1^2-2x_1x_2+tx^2_{2}\).

    For which \(t\) does \(Q_t\) have rank \(1\)? [4]

Section B

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

    • (a) Let \(\pi _1,\pi _2,\pi _3\in L(V)\) satisfy

      \begin{align*} \pi _i\circ \pi _j&=\delta _{ij}\pi _i, \quad \text {for $1\leq i,j\leq 3$}\\ \mathrm {id}_V&=\pi _1+\pi _2+\pi _3. \end{align*}

      • (i) Show that \(V=\im \pi _1\oplus \im \pi _{2}\oplus \im \pi _{3}\).

      • (ii) Show that \(\ker \pi _1=\im \pi _2\oplus \im \pi _3\).


    • (b) Let \(\phi \in L(V)\) be a linear operator on \(V\) and \(U\leq V\) a \(\phi \)-invariant subspace. Let \(q:V\to V/U\) be the quotient map.

      Show that there is a well-defined linear operator \(\tilde {\phi }\in L(V/U)\) such that

      \begin{equation*} \tilde {\phi }(q(v))=q(\phi (v)), \end{equation*}

      for \(v\in V\). [6]

  • 8.

    • (a) Let \(V\) be a finite-dimensional complex inner product space and \(\phi \in L(V)\) a linear operator.

      Show that \(\lambda \) is an eigenvalue of \(\phi \) if and only if \(\bar {\lambda }\) is an eigenvalue of \(\phi ^{*}\). [6]

    • (b) Let \(U=\Span {(1,1,-1,-1),(1,0,0,-1)}\leq \R ^4\) and view \(\R ^4\) as an inner product space using dot product.

      • (i) Compute \(U^{\perp }\).

      • (ii) Find an orthonormal basis of \(U^{\perp }\).

      • (iii) Compute the orthogonal projections of \((1,2,3,1)\) onto \(U\) and \(U^{\perp }\).


  • 9.

    • (a) Let \(V\) be a finite-dimensional real vector space and \(\alpha ,\beta \in V^{*}\) linearly independent linear functionals.

      Define a symmetric bilinear form \(B\) on \(V\) by

      \begin{equation*} B(v,w)=\tfrac 12(\alpha (v)\beta (w)+\alpha (w)\beta (v)). \end{equation*}

      Compute the rank and signature of \(B\). [6]

    • (b) Let \(t\in \R \) and define \(A_{t}\) by

      \begin{equation*} A_t= \begin{pmatrix} 1&2&0\\2&0&t\\0&t&3 \end {pmatrix}. \end{equation*}

      For which \(t\) does the symmetric bilinear form \(B_{A_t}\) have signature \((3,0)\)? [12]