| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
38 |
|
Paper 2, Section II, E |
2021 |
(a) Compute the characteristic polynomial and minimal polynomial of
$$A=\left(\begin{array}{ccc}
-2 & -6 & -9 \\
3 & 7 & 9 \\
-1 & -2 & -2
\end{array}\right)$$
Write down the Jordan normal form for $A$.
(b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}, f: V \rightarrow V$ be a linear map, and for $\alpha \in \mathbb{C}, n \geqslant 1$, write
$$W_{\alpha, n}:=\left{v \in V \mid(f-\alpha I)^{n} v=0\right}$$
(i) Given $v \in W_{\alpha, n}, v \neq 0$, construct a non-zero eigenvector for $f$ in terms of $v$.
(ii) Show that if $w_{1}, \ldots, w_{d}$ are non-zero eigenvectors for $f$ with eigenvalues $\alpha_{1}, \ldots, \alpha_{d}$, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $w_{1}, \ldots, w_{d}$ are linearly independent.
(iii) Show that if $v_{1} \in W_{\alpha_{1}, n}, \ldots, v_{d} \in W_{\alpha_{d}, n}$ are all non-zero, and $\alpha_{i} \neq \alpha_{j}$ for all $i \neq j$, then $v_{1}, \ldots, v_{d}$ are linearly independent.