# Be an eigenvector of an invertible matrix a which of

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be an eigenvector of an invertible matrix A . Which of the following are necessarily true? Please give your reasoning. I. ~v is an eigenvector of A - 1 . II. ~v is an eigenvector of A 2 . III. ~v is an eigenvector of A + I .

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Name (print) 2 A–3. True or false? If false, give a reason. a) If { ~v 1 ,~v 2 ,~v 3 } is a collection of non-zero vectors in R 5 , then the span of { ~v 1 ,~v 2 ,~v 3 } must be a three-dimensional subspace of R 5 . b) The set of polynomials in P 4 satisfying p (0) = 2 is a subspace of P 4 . c) If ~x is a least-squares solution to A~x = ~ b , then A~x is orthogonal to the image of A . d) If ~v 1 ,~v 2 ,~v 3 are orthonormal vectors in R 3 , then these vectors are linearly independent. e) If the matrix A is both invertible and diagonalizable, then A - 1 is diagonalizable. A–4. Consider the matrix A = 2 - 1 2 2 . If ~x R 2 is a unit vector, what is the largest that k A~x k could possibly be?
Name (print) 3 A–5. Let A be an m × n matrix, and suppose ~v and ~w are orthogonal eigenvectors of A T A . Show that A~v and A~w are orthogonal.

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