Exercise 6.7.10

Answers

Let $β^{'}$ and $γ^{'}$ be the standard bases for $F^{m}$ and $F^{n}$ respectively. Thus we have ${[L_{A}]}_{β^{'}}^{γ^{'}} = A$ . Also, let

β = {v_{1}, v_{2}, \dots, v_{m}}

and

γ = {u_{1}, u_{2}, \dots, u_{n}} .

By Theorem 6.26 we know that ${[L_{A}]}_{β}^{γ} = Σ$ . Apply Exercise 2.5.8 we have

A = {[L_{A}]}_{β^{'}}^{γ^{'}} = {[I]}_{γ}^{γ^{'}} {[L_{A}]}_{β}^{γ} {[I]}_{β^{'}}^{β} = U Σ V^{∗} .

jephianlin

2011-06-27 00:00