Exercise 2.1.35

Answers

1.
With the hypothesis V = R(T) + N(T), it’s sufficient to say that R(T) ∩ N(T) = {0}. But this is easy since
dim(R(T) ∩ N(T)) = dim(R(T)) + dim(N(T)) −dim(R(T) + N(T))
= dim(R(T)) + dim(N(T)) −dim(V ) = 0

by Dimension Theorem.

2.
Similarly we have
dim(R(T) + N(T)) = dim(R(T)) + dim(N(T)) −dim(R(T) ∩ N(T))
= dim(R(T)) + dim(N(T)) −dim({0}) = dim(V )

by Dimension Theorem. So we have V = R(T) + N(T).

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2011-06-27 00:00
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