Linear Algebra

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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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