Let A be an n × n matrix. Show that a vector x in either Rn or Cn is an eigenvector belonging to A if and only if the subspace S spanned by x and Ax has dimension 1

Find a basis for the subspace W = {(a,b,c) ∈ R3 | b = 2a−3c}.

Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.

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.

Find an orthonormal basis of the subspace spanned by the vectors in Exercise 3.