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Unformatted text preview: xists a basis ﬁ for such that
[T]ﬁ is diagonal. By (a), [T−1 ]ﬁ must then also be diagonal.
5.2.20 Suppose that
= W1 ⊕ · · · ⊕
. Then, by Theorem 5.10, we can
ﬁnd ordered bases ﬁ1
k for W1
k , respectively, so that ﬁ =
ﬁ1 ∪ · · · ∪ is a basis for . This then gives that dim( ) = dim(W1 ) +
· · · + dimWk by counting basis vectors.
On the other hand, suppose that the dimension equality holds. Picking
a basis ﬁi for each Wi, we have that ﬁ = ﬁ1 ∪· · ·∪ has dim( ) vectors.
Since W1 + · · · + Wk = , we also have that ﬁ...
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- Spring '10