Exercise 6.9.3

Answers

1.: The set ${w_{1}, w_{2}}$ is linearly independent by definition. Also, we have $w_{1} = e_{1} + e_{4}$ and $w_{2} = e_{1} - e_{4}$ are both elements in $span ({e_{1}, e_{4}})$ . Hence it’s a basis for $span ({e_{1}, e_{4}})$ . Naturally, it’s orthogonal since $⟨ w_{1}, w_{2} ⟩ = 0$ .
2.: For brevity, we write $W = span ({e_{1}, e_{4}})$ . We have $T_{v} (W) \subset W$ and $T_{v}^{∗} (W) \subset W$ by Theorem 6.39. Also, $W$ is $L_{A}$ -invariant if we directly check it. Hence $W$ is $T_{v}^{∗} L_{A} T_{v}$ -invariant.

jephianlin

2011-06-27 00:00