This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 finitedimensional 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...
View
Full
Document
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.
 Spring '08
 GUREVITCH
 Math, Linear Algebra, Algebra

Click to edit the document details