Exercise 2.3.1

Answers

1.: It should be ${[UT]}_{α}^{γ} = {[T]}_{α}^{β} {[U]}_{β}^{γ}$ .
2.: Yes. That’s Theorem 2.14.
3.: No. In general $β$ is not a basis for $V$ .
4.: Yes. That’s Theorem 2.12.
5.: No. It will be true when $β = α$ .
6.: No. We have ${(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})}^{2} = I$ .
7.: No. $T$ is a transformation from $V$ to $W$ but $L_{A}$ can only be a transformation from $𝔽^{m}$ to $𝔽^{n}$ .
8.: No. We have ${(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})}^{2} = I$ .
9.: Yes. That’s Theorem 2.15.
10.: Yes. Since $δ_{ij} = 1$ only when $i = j$ , we have $A_{ij} = δ_{i} j$ .

jephianlin

2011-06-27 00:00