exam-23

Section A

1. Let \(U,W\leq \R ^3\) be given by \(U=\Span {(1,1,1), (1,2,1)}\), \(W=\Span {(1,0,0),(0,0,1)}\).

Write down bases for \(U\cap W\) and \(U+W\). [4]

2. Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space \(V\) such that
\(\seteqnumber{0}{}{0}\)
\begin{equation*} \ker \phi \cap \im \phi =\set {0}. \end{equation*}

Prove that \(\ker \phi \oplus \im \phi =V\). [4]

3. Find the minimal polynomial of \(A\in M_3(\R )\) given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*} A= \begin{pmatrix} 0&1&1\\0&0&2\\1&0&0 \end {pmatrix}. \end{equation*}

[4]

4. Let \(\phi \in L(V)\) be a linear operator on a complex vector space \(V\) with characteristic polynomial \((x-3)^{4}(x-2)^{2}\) and minimal polynomial \((x-3)^{2}(x-2)\).

What are the possible Jordan normal forms of \(\phi \)? [4]

5. Let \(E\leq (\R ^{3})^{*}\) be spanned by \(\alpha \) given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*} \alpha (x)=x_1+2x_2-x_3. \end{equation*}

Write down a basis of \(\sol E\). [4]

6. Define the rank and signature of a symmetric bilinear form \(B\) on a real, finite-dimensional vector space.

State Sylvester’s Law of Inertia. [4]

Section B

7. Let \(V\) be a vector space over a field \(\F \) and \(\phi \in L(V)\) a linear operator on \(V\).
- (a) Suppose that \(V=V_1\oplus V_2\) with each \(V_i\) \(\phi \)-invariant. Show that
  \(\seteqnumber{0}{}{0}\)
  \begin{equation*} \ker \phi =\ker \phi \restr {V_1}\oplus \ker \phi \restr {V_2}. \end{equation*}
  
  [6]
- (b) Let \(U\leq V\) be \(\phi \)-invariant and let \(q:V\to V/U\) be the quotient map.
  - (i) Show that there is a well-defined linear operator \(\bar {\phi }\) on \(V/U\) such that
    \(\seteqnumber{0}{}{0}\)
    \begin{equation*} \bar {\phi }(q(v))=q(\phi (v)), \end{equation*}
    
    for all \(v\in V\).
  - (ii) If \(\dim V=n\), \(u_1,\dots ,u_k\) is a basis of \(U\) and \(q(v_{1}),\dots ,q(v_{n-k})\) is a basis of \(V/U\), show that \(u_1,\dots ,u_k,v_1,\dots ,v_{n-k}\) is a basis of \(V\).
[12]

8. Let \(A\) be given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*} A= \begin{pmatrix*}[r] 1&-1&2\\-3&-1&-3\\1&1&0 \end {pmatrix*}. \end{equation*}

The characteristic polynomial of \(A\) is \((2-x)(x+1)^{2}\) (you do not need to prove this).
- (a) Compute the minimum polynomial of \(A\). [5]
- (b) Find the Jordan normal form of \(A\). [4]
- (c) Find a Jordan basis for \(A\). [9]

9.
- (a) Let \(q:\R ^5\to \R \) be a quadratic form. Which of the following are possible signatures of \(q\)?
  - (i) \((4,2)\).
  - (ii) \((2,-1)\).
  - (iii) \((2,1)\).
  - (iv) \((2,3)\).
  In each case, briefly justify your answer. [8]
- (b) Let \(t\in \R \) and define \(A_t\) by
  \(\seteqnumber{0}{}{0}\)
  \begin{equation*} A_t= \begin{pmatrix*}[r] 1&0&-1\\0&2&t\\-1&t&0 \end {pmatrix*}. \end{equation*}
  
  Find an invertible matrix \(P_t\) such that \(P_t^TA_tP_t\) is diagonal. [10]