Unformatted text preview: 3.5. LINEAR ALGEBRA OVER Z 185 3.4.13. Let V , W be finite–dimensional vector spaces over K . Let B , B be two ordered bases of V , and let C , C be two ordered bases of W . Write F D OE id Ł C ;C and G D OE id Ł B ;B . Let T 2 Hom K .V;W / and S 2 End K .V / . Show that OETŁ C ;B D F OETŁ C;B G 1 . 3.4.14. Suppose that T and T are two linear transformations of a finite dimensional vector space V , and that B and B are two ordered bases of V . Show that OETŁ B and OET Ł B are similar matrices if, and only if, T and T are similar linear transformations. 3.4.15. Let T be the linear transformation of R 3 with standard matrix 2 4 1 5 2 2 1 3 1 1 4 3 5 . Find the matrix of OETŁ B of T with respect to the ordered basis B D @ 2 4 1 1 1 3 5 ; 2 4 1 1 3 5 ; 2 4 1 3 5 1 A . 3.4.16. Show that 1 1 0 1 is not similar to any matrix of the form a 0 0 b . (Hint: Suppose the two matrices are similar. Use the similarity invariants determinant and trace to derive information about a and b .) 3.4.17.3....
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, Abelian group, @415, ﬁnite–dimensional vector spaces

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