Exercise 6.3.9

Answers

For each vector $v \in V$ we may write $v = v_{1} + v_{2}$ such that $v_{1} \in W$ and $v_{2} \in W^{⊥}$ . Now check

⟨ x_{1} + x_{2}, T^{∗} (y_{1} + y_{2}) ⟩ = ⟨ T (x_{1} + x_{2}), y_{1} + y_{2} ⟩

= ⟨ x_{1}, y_{1} + y_{2} ⟩ = ⟨ x_{1}, y_{1} ⟩

and

⟨ x_{1} + x_{2}, T (y_{1} + y_{2}) ⟩ = ⟨ x_{1} + x_{2}, y_{1} ⟩

= ⟨ x_{1}, y_{1} ⟩ = ⟨ x_{1} + x_{2}, T^{∗} (y_{1} + y_{2}) ⟩

for all $x = x_{1} + x_{2}$ and $y = y_{1} + y_{2}$ .

jephianlin

2011-06-27 00:00