Exercise 6.11.16

Answers

(c)

As the definition of $T_{i}$ given in the Corollary, we know that

T_{i} T_{j} (x) = \dots + T (x_{i}) + \dots + T (x_{j}) + \dots T_{j} T_{i} (x),

where

x = x_{1} + x_{2} + \dots + x_{m} .

(d)

Again, we have

T (x) = T (x_{1}) + T (x_{2}) + \dots + T (x_{m}) = T_{1} T_{2} \dots T_{m} (x),

where

x = x_{1} + x_{2} + \dots + x_{m} .

(e)

We know that $\det (T_{W_{i}}) = \det (T_{i})$ since $V = W_{i} \oplus W_{i}^{⊥}$ and

\det (T_{W_{i}^{⊥}}) = \det (I_{W_{i}^{⊥}}) = 1 .

So $T_{i}$ is a rotation if and only if $T_{W_{i}}$ is a rotation. By Theorem 6.47 we get the result.

jephianlin

2011-06-27 00:00