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Linear Algebra

Exercise 6.11.14

Answers

1.

It comes from that

∥ T_{W} (x) ∥ = ∥ T (x) ∥ = ∥ x ∥ .

2.

Suppose

y

is an element in

W^{⊥}

. Since

T_{W}

is invertible by the previous argument, for each

x \in W

we have

x = T (z)

for some

z \in W

. This means that

⟨ T (y), x ⟩ = ⟨ y, T^{∗} T (z) ⟩ = ⟨ y, z ⟩ = 0 .

3.

It comes from that

∥ T_{W^{⊥}} (x) ∥ = ∥ T (x) ∥ = ∥ x ∥ .

jephianlin

2011-06-27 00:00

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