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# Linear Algebra

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1 Math 110 Homework 11 Partial Solutions If you have any questions about these solutions, or about any problem not solved, please ask via email or in office hours, etc. 5.2.12 (a) Let E λ be the eigenspace of T corresponding to λ , and let F λ - 1 be the eigenspace of T - 1 corresponding to λ - 1 . If x E λ , then T ( x ) = λx . Thus x = T - 1 ( λx ) = λT - 1 ( x ) and so T - 1 ( x ) = λ - 1 x . Thus x F λ - 1 . By the same argument, we also get F λ - 1 E λ and thus that they are equal. (b) If T is diagonalizable, then there exists a basis β for V such that [ T ] fi is diagonal. By (a), [ T - 1 ] fi must then also be diagonal. 5.2.20 Suppose that V = W 1 ⊕ · · · ⊕ W k . Then, by Theorem 5.10, we can find ordered bases β 1 , . . . , β k for W 1 , . . . , W k , respectively, so that β = β 1 ∪· · ·∪ β k is a basis for V . This then gives that dim( V ) = dim( W 1 )+ · · · + dimW k by counting basis vectors. On the other hand, suppose that the dimension equality holds. Picking a basis β i for each W i , we have that β = β 1 ∪· · ·∪ β k has dim( V ) vectors.

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5.2+5.4 - 1 Math 110 Homework 11 Partial Solutions If you...

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