Exercise 5.4.1

Answers

1.: No. The subspace ${0}$ must be a $T$ -invariant subspace.
2.: Yes. This is Theorem 5.21.
3.: No. For example, let $T$ be the identity map from $ℝ$ to $ℝ$ and $v = (1)$ and $w = (2)$ . Then we have $W = W^{'} = ℝ$ .
4.: No. For example, let $T$ be the mapping from $ℝ^{2}$ to $ℝ^{2}$ defined by $T (x, y) = y$ . Pick $v = (1, 1)$ . Thus the $T$ -cyclic subspace generated by $v$ and $T (v)$ are $ℝ^{2}$ and the $x$ -axis.
5.: Yes. The characteristic polynomial is the described polynomial.
6.: Yes. We may prove it by induction or just use Exercise 5.1.21.
7.: Yes. This is Theorem 5.25.

jephianlin

2011-06-27 00:00