Section A

  • 1. Which of the following subsets of \(\R ^3\) are linear subspaces? In each case, briefly justify your answer.

    • (a) \(U_1:=\set {(x_1,x_2,x_3)\in \R ^3\st x_1^2=x_2^2}\);

    • (b) \(U_{2}:=\Z ^3=\set {(x_1,x_2,x_3)\in \R ^3\st x_1,x_2,x_3\in \Z }\);

    • (c) \(U_3:=\set {(x_1,x_2,x_3)\in \R ^3\st (x_1-x_3)^2+x_2^2=0}\).

    [4]

  • 2. Let \(U=\Span {(1,2,1)}\leq \R ^3\). Which of the following equalities hold in the quotient space \(\R ^3/U\)? In each case, briefly justify your answer.

    • (a) \((1,2,3)+U=(4,8,7)+U\);

    • (b) \(((1,1,2)+U)+((2,7,2)+U)=(-2,-2,-1)+U\).

    [4]

  • 3. Compute the minimum polynomial and Jordan normal form of the matrix \(A\in M_2(\C )\) given by

    \begin{equation*} A= \begin{pmatrix} -1&1\\-4&3 \end {pmatrix}. \end{equation*}

    [4]

  • 4. With \(A\) as in question 3, compute \(A^4-4A^3+6A^2-4A+I_2\), where \(I_2\) is the \(2\times 2\) identity matrix. [4]

  • 5. For each pair \((p,q)\) below, either give an example of a quadratic form on \(\R ^5\) with signature \((p,q)\) or explain why one does not exist.

    • (i) \((3,2)\).

    • (ii) \((2,0)\).

    • (iii) \((3,3)\).

    • (iv) \((-2,2)\).

    [4]

Section B

  • 6.

    • (a) Let \(V\) be a finite-dimensional vector space over a field \(\F \) and let \(U_1,U_2,U_3\leq V\) be subspaces.

      Suppose that

      \begin{equation*} \dim (U_1+U_2+U_3)=\dim U_1+\dim U_2+\dim U_3. \end{equation*}

      Show that the sum \(U_1+U_2+U_3\) is direct.

      [State any results from lectures that you use.] [6]

    • (b) Let \(A\) be given by

      \begin{equation*} A=\begin{pmatrix} -1&1&0\\-3&3&-1\\-2&1&1 \end {pmatrix}. \end{equation*}

      • (i) Compute the characteristic and minimum polynomials of \(A\).

      • (ii) Find the Jordan normal form of \(A\).

      • (iii) Find a Jordan basis for \(A\).

      [9]

  • 7.

    • (a) Let \(A\) be a square matrix with characteristic polynomial \((x-2)^4(x-1)^2\) and minimal polynomial \((x-2)^2(x-1)\). What are the possibilities for the Jordan normal form of \(A\)? [6]

    • (b) Let \(t\in \R \) and define a quadratic form \(q_t:\R ^3\to \R \) by

      \begin{equation*} q_t(x)=x_1^2+2x_2^2-x_3^2+2tx_1x_3+4x_1x_2. \end{equation*}

      What is the rank and signature of \(q_t\)? (Your answer will depend on \(t\).) [9]