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Exercise 6.8.13

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1.

x

is an element in

V

, then

ϕ_{γ} (x)

and

ϕ_{β} (x)

are the

γ

-coordinates and the

β

-coordinates respectively. By the definition of

Q

, we have

ϕ_{β} (x) = L_{Q} ϕ_{γ} (x)

for all $x \in V$ .

2.

By the Corollary 2 after Theorem 6.32, we know that

H (x, y) = {[ϕ_{γ} (x)]}^{t} ψ_{γ} (H) [ϕ_{γ} (y)] = {[ϕ_{β} (x)]}^{t} ψ_{β} (H) [ϕ_{β} (y)] .

By the previous argument we know that

{[ϕ_{γ} (x)]}^{t} Q^{t} ψ_{β} (H) Q [ϕ_{γ} (y)] = {[ϕ_{γ} (x)]}^{t} ψ_{γ} (H) [ϕ_{γ} (y)],

where $Q$ is the change of coordinate matrix changing $γ$ -coordinates to $β$ -coordinates. Again, by the Corollary 2 after Theorem 6.32 we know the matrix $Q^{t} ψ_{β} (H) Q$ must be the matrix $ψ_{γ} (H)$ . Hence they are congruent.

jephianlin

2011-06-27 00:00

Exercise 6.8.13

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