Exercise 6.1.6

Answers

1.
⟨x,y + z⟩ = ⟨y + z,x⟩¯ = ⟨y,x⟩¯ + ⟨z,x⟩¯ = ⟨x,y⟩ + ⟨x,z⟩.
2.
⟨x,cy⟩ = ⟨cy,x⟩¯ = c ⟨y,x⟩¯ = c¯ ⟨x,y⟩.
3.
⟨x,0⟩ = 0¯ ⟨x,0⟩ = 0

and

⟨0,x⟩ = ⟨x,0⟩¯ = 0.
4.
If x = 0, then ⟨0,0⟩ = 0 by previous rule. If x≠0, then ⟨x,x⟩ > 0.
5.
If ⟨x,y⟩ = ⟨x,z⟩ for all x ∈ V , we have ⟨x,y − z⟩ = 0 for all x ∈ V . So we have ⟨y − z,y − z⟩ = 0 and hence y − z = 0.
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2011-06-27 00:00
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