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6 Nov 2019
Math 308 Proof Homework 4 (Advanced Linear Algebra)
I am not very good at Proofs and need some help, please.
1. Suppose T Rn R n is a linear map that satisfies T2 TT. or equivalently (T) (r) for all E R" Note: You may NOT assume T is invertible. In fact, any projection map T satisfies T IT, and I is the only invertible one. This problem will show that T T implies T is a projection map (a) Show that ker(T) and range(T) are complements in R". (b) Show that TT is the projection onto range with respect to ker(T). (c) Show that if A is eigenvalue of T, then A 0 or 1. 2. Suppose that (A, iT is an eigenpair of matrix A. (a) Show (A U is an eigenpair of A*, for all positive integers k. (b) Assume A is invertible. Why does this mean A 0? Show (A-1, i is an eigenpair of A 3. Suppose that tvi win is a basis of R" which are eigenvectors of both A and B. That is, say (i) (A1, vi), (An, min) are eigenpairs of A (ii) (pi, vi), pra, vn) are eigenpairs of B Show that AB BA Show transcribed image text
Math 308 Proof Homework 4 (Advanced Linear Algebra)
I am not very good at Proofs and need some help, please.
1. Suppose T Rn R n is a linear map that satisfies T2 TT. or equivalently (T) (r) for all E R" Note: You may NOT assume T is invertible. In fact, any projection map T satisfies T IT, and I is the only invertible one. This problem will show that T T implies T is a projection map (a) Show that ker(T) and range(T) are complements in R". (b) Show that TT is the projection onto range with respect to ker(T). (c) Show that if A is eigenvalue of T, then A 0 or 1. 2. Suppose that (A, iT is an eigenpair of matrix A. (a) Show (A U is an eigenpair of A*, for all positive integers k. (b) Assume A is invertible. Why does this mean A 0? Show (A-1, i is an eigenpair of A 3. Suppose that tvi win is a basis of R" which are eigenvectors of both A and B. That is, say (i) (A1, vi), (An, min) are eigenpairs of A (ii) (pi, vi), pra, vn) are eigenpairs of B Show that AB BA
Show transcribed image text Sixta KovacekLv2
27 May 2019