{(: ) : (d) : a, b, cE R. C

Let U and V be subspaces of Rn. a) Show that Un V = {√ € R¹ : √ € U and ʊ € V} is a subspace of R". b) Let U = null(A) and V = null(B), where A, B are matrices with n columns. Express UV as either null(C) or im(C) for some matrix C. (You may wish to write C as a block matrix.) c) Let U = null(X) where X has n columns, and V = im(Y), where Y has n rows. Show that if UnV ‡ {0} then XY is not invertible.

Given an arbitrary m x n matrix M and a vector b0, is the solution set of Mv = b a vector space? Explain.

Let V be a vector space of dimension n, and let B and C be bases for V. Then the product PB- cPc-B is equal to the n × n identity matrix. True False

Let A and B be m x n matrices. (a) Suppose that v"Aw A = B. v" Bw for all vectors v, w with real entries. Prove that