Exercise 6.7.9

Answers

1.

Use the fact that

⟨ T^{∗} (u_{i}), v_{j} ⟩ = {\begin{matrix} ⟨ u_{i}, σ_{j} u_{j} ⟩ & if & j \leq r; \\ ⟨ u_{i}, 0 ⟩ & if & j > r, \end{matrix}

= δ_{ij} σ_{j}

for $i \leq r$ . We know that $T^{∗} (u_{i}) = σ_{i} v_{i}$ . And so

T T^{∗} (u_{i}) = T (σ_{i} v_{i}) = σ^{2} v_{i}

for $j \leq r$ . Similarly, we know that, for $i > r$ ,

⟨ T^{∗} (u_{i}), v_{j} ⟩ = 0 .

Hence $T T^{∗} (u_{i}) = 0$ when $i > r$ .

2.

Let

T = L_{A}

and use the previous argument.

3.

Use Theorem 6.26 and Exercise 6.7.9(a).

4.

Replace

T

A

jephianlin

2011-06-27 00:00