math110s-hw5 - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

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Unformatted text preview: MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 5 If T ¢ V V is a linear operator, we will write T n n times ³¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹µ T X X T , ie. T composed with itself n times. For example, T 2 T X T , T 3 T X T X T , etc. We will let O ¢ V W denote the zero linear transformation, ie. O v W for all v > V . The set of all linear transformations from V to W is denoted Hom F V,W . We write End F V for Hom F V,V . 1. Let U,W,V be finite-dimensional vector spaces over F . Let α,β > F . (a) Let T ¢ V W be a linear transformation. Show that rank T B dim V . (b) Let S ¢ U V and T ¢ V W be linear transformations. Show that rank T X S B rank T and rank T X S B rank S . (c) Let S 1 ¢ U V , S 2 ¢ U V , and T ¢ V W be linear transformations. Show that T X α S 1 β S 2 α T X S 1 β T X S 2 . (d) Let S ¢ U V , T 1 ¢ V W , and T 2 ¢ V W be linear transformations. Show that α T 1 β T 2 X S α T 1 X...
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This note was uploaded on 08/01/2008 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at University of California, Berkeley.

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math110s-hw5 - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

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