Exercise 5.4.27

Answers

1.
If v + W = v′ + W, we know that v − v′ is an element in W by Exercise 1.3.31. Since W is T-invariant, we have
T(v) − T(v′) = T(v − v′)

is in W. So we have

T(v) + W = T(v′) + W

and this means

T¯(v + W) = T¯(v′ + W).
2.
Just check that
T¯((v + v′) + W) = T(v + v′) + W
= (T(v) + W) + (T(v′) + W) = T¯(v + W) + T¯(v′ + W)

and

T¯(cv + W) = T(cv) + W = c(T(v) + W) = cT¯(v + W).
3.
For each v ∈ V we might see
ηT(v) = T(v) + W = T¯(v + W) = T¯η(v).
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2011-06-27 00:00
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