Let A =  a b c d , B =  e f g h ∈ M2,2(R) and define hA, Bi = ae + bf + 2cg + 2dh. (a) Show hA, Bi is an inner product of M2,2(R). (b) Find an orthonormal basis for the subspace U of M2,2(R) spanned by −1 0 1 1 ,  0 1 1 0 .

(a) To show that hA, Bi is an inner product on M2,2(R), we need to verify the four properties of an inner product:

1. Conjugate Symmetry: hA, Bi = hB, Ai
2. Linearity in the first argument: hλA + μB, Ci = λhA, Ci + μhB, Ci for all λ, μ ∈ R and all A, B, C ∈ M2,2(R)
3. Positive-definiteness: hA, Ai ≥ 0 with equality if and only if A = 0
4. Non-degeneracy: If hA, Bi = 0 for all B ∈ M2,2(R), then A = 0

Let's verify these properties:

1. Conjugate Symmetry: hA, Bi = ae + bf + 2cg + 2dh = hB, Ai
2. Linearity in the first argument: hλA + μB, Ci = λ(ae + bf + 2cg + 2dh) + μ(ae + bf + 2cg + 2dh) = λhA, Ci + μhB, Ci
3. Positive-definiteness: hA, Ai = a^2 + b^2 + 2c^2 + 2d^2 ≥ 0 with equality if and only if a = b = c = d = 0, i.e., A = 0
4. Non-degeneracy: If hA, Bi = 0 for all B ∈ M2,2(R), then ae + bf + 2cg + 2dh = 0 for all e, f, g, h ∈ R, which implies a = b = c = d = 0, i.e., A = 0

So, hA, Bi is an inner product on M2,2(R).

(b) The subspace U of M2,2(R) is spanned by the matrices A = [-1 0; 1 1] and B = [0 1; 1 0]. To find an orthonormal basis, we first need to find an orthogonal basis using the Gram-Schmidt process, and then normalize the basis vectors.

Let's start with A as our first basis vector. Then, we subtract the projection of B onto A from B to get our second basis vector:

B' = B - (hB, Ai / hA, Ai) * A

Then, we normalize the basis vectors:

A' = A / ||A|| B'' = B' / ||B'||

These normalized vectors A' and B'' form an orthonormal basis for U.