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Unformatted text preview: 11) Let T be the linear operator on F 2 which is represented in the standard ordered basis by the matrix 0 0 1 0 . Let a 1 = (0 , 1). Show that F 2 6 = Z ( a 1 ,T ) and that there is no non-zero vector a 2 in F 2 such that Z ( a 2 ,T ) is disjoint from Z ( a 1 ,T ). 11) Let T be a linear operator on V an n-dimensional vectorspace and let R = T ( V ) be the range of T . (a) Prove that R has a complementary T-invariant subspace iff R is independent of N = null T . (b) If R and N are independent, prove that N is the unique T-invariant subspace complementary to R . 11) Let T be the linear operator on R 3 which is represented by the matrix 2 0 0 1 2 0 0 0 3 . Let W be the null space of T- 2 I . Prove that W has no complementary T-invariant subspace. 11) Let T be the linear operator on F 4 which is represented by the matrix c 0 0 0 1 c 0 0 0 1 c 0 0 1 c and let W be the nullspace of T- cI ....
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This document was uploaded on 01/25/2012.
- Spring '09