Exercise 6.6.11

Answers

By Theorem 6.25(a), we may uniquely write element in the space as

v = x_{1} + x_{2} + \dots + x_{k}

such that $x_{i} \in W_{i}$ . If $i \neq j$ , we have

T_{i} T_{j} (v) = 0 = δ_{ij} T_{i} (v)

by the definition of $T_{i}$ ’s and Theorem 6.25(b). Similarly, if $i = j$ , we have

T_{i} T_{i} (v) = x_{i} = δ_{ii} T_{i} (v) .

So they are the same.

jephianlin

2011-06-27 00:00