MA22020: Exercise sheet 4

Warmup questions

1. Write down matrices \(A\in M_n(\R )\) of the following forms:
- (a) \(A_1\oplus A_2\oplus A_3\) with each \(A_i\in M_2(\R )\).
- (b) \(\oplst {A}15\) with each \(A_i\in M_1(\R )\).
- (c) \(A\in M_3(\R )\) such that \(A\) is not of the form \(\oplst {A}1k\) with \(k>1\).
2. Let \(\lst {V}1k\leq V\) and \(\phi _i\in L(V_i)\), \(\bw 1ik\). Suppose that \(V=\oplst {V}1k\) and set \(\phi =\oplst \phi 1k\).
- (a) If \(U_i\leq V_i\), \(\bw 1ik\), show that the sum \(\plst {U}1k\) is direct.
- (b) Prove that \(\im \phi =\oplst {\im \phi }1k\).
3. In the situation of Question 2, prove:
- (a) \(m_{\phi _i}\) divides \(m_{\phi }\), for each \(\bw 1ik\).
- (b) If each \(m_{\phi _i}\) divides \(p\in \F [x]\), then \(p(\phi )=0\).
Thus \(m_{\phi }\) is the monic polynomial of smallest degree divided by each \(m_{\phi _i}\). Otherwise said, \(m_{\phi }\) is the least common multiple of \(m_{\phi _1},\dots ,m_{\phi _k}\).
4. Let \(\phi =\phi _A\in L(\C ^3)\) where \(A\) is given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*} \begin{pmatrix*} 0&0&0\\4&0&0\\0&0&5 \end {pmatrix*}. \end{equation*}
- (a) Compute the characteristic and minimum polynomials of \(\phi \).
- (b) Find bases for the eigenspaces and generalised eigenspaces of \(\phi \).
Homework questions
5. Let \(\phi \in L(V)\) be a linear operator on a vector space \(V\).

Prove that \(\im \phi ^{k}\geq \im \phi ^{k+1}\), for all \(k\in \N \). Moreover, if \(\im \phi ^{k}=\im \phi ^{k+1}\) then \(\im \phi ^k=\im \phi ^{k+n}\), for all \(n\in \N \).
6. Let \(\phi =\phi _A\in L(\C ^3)\) where \(A\) is given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*} \begin{pmatrix*}[r] 0&1&-1\\-10&-2&5\\-6&2&1 \end {pmatrix*}. \end{equation*}
- (a) Compute the characteristic and minimum polynomials of \(\phi \).
- (b) Find bases for the eigenspaces and generalised eigenspaces of \(\phi \).

Please hand in at 4W level 1 by NOON on Thursday 28th November

MA22020: Exercise sheet 4

Warmup questions

Homework questions