1. Let V = M2×2(R), and let T
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
. 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
(c) Compute the Fourier coecients of x = (1, 1, 2) relative to β
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