Linear Algebra Problems

1. Let V = M2×2(R), and let T  a b c d =  d b c a be a linear operator on V . Find the eigenvalues of T, and an ordered basis β for V such that [T]β is a diagonal matrix. 2. Prove that the eigenvalues of an upper triangular matrix A are the diagonal entries. 3. Prove that similar matrices have the same characteristic polynomial. 4. Two matrices A, B ∈ M2×2(R) are simultaneously diagonalizable if there exists an invertible matrix Q ∈ M2×2 such that both Q−1AQ and Q−1BQ are diagonal matrices. Prove that if A and B are simultaneously digonalizable matrices, then A and B commute (i.e. AB = BA). 5. Let T be a linear operator on an inner product space V , and suppose that ||T(x)|| = ||x|| for all x. Prove that T is one-to-one. 6. Consider the inner product space V = R 3 . Let S = (1, 0, 1),(0, 1, 1),(1, 3, 3). (a) Use the Gram-Schmidt process on S to obtain an orthogonal basis β for Span(S). (b) Normalize the vectors in β to obtain an orthonormal basis β 0 for S. (c) Compute the Fourier coecients of x = (1, 1, 2) relative to β