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

First, let's define the two given matrices:

Matrix 1 (M1) = [ -1 0 ] [ 1 1 ]

Matrix 2 (M2) = [ 0 1 ] [ 1 0 ]

The subspace U is spanned by these two matrices, meaning any matrix in U can be written as a linear combination of M1 and M2.

To find an orthonormal basis, we first need to check if these two matrices are orthogonal to each other. Two matrices are orthogonal if their dot product is zero. The dot product of two matrices is calculated by multiplying corresponding entries and then summing up all those products.

The dot product of M1 and M2 using the given definition (A, B) = ae + bf + 2cg + 2dh is:

(-10 + 01 + 211 + 210) = 2

Since the dot product is not zero, M1 and M2 are not orthogonal.

To find an orthonormal basis, we need to make these two matrices orthogonal and normalize them.

We can use the Gram-Schmidt process to orthogonalize the matrices:

1. Let V1 = M1 = [ -1 0 ] [ 1 1 ]

2. Let V2 = M2 - proj_V1(M2)

To find proj_V1(M2), we need to find the dot product of M2 and V1, and then divide by the dot product of V1 and V1:

proj_V1(M2) = ((M2, V1) / (V1, V1)) * V1 = ((0*-1 + 10 + 210 + 211) / (-1-1 + 00 + 211 + 21*0)) * V1 = (2 / 3) * V1 = [ -2/3 0 ] [ 2/3 2/3 ]

So, V2 = M2 - proj_V1(M2) = [ 0 1 ] - [ -2/3 0 ] [ 1 0 ] [ 2/3 2/3 ] = [ 2/3 1 ] [ 1/3 -2/3 ]

Now, we have two orthogonal matrices V1 and V2.

To normalize them, we divide each matrix by its norm. The norm of a matrix is the square root of the sum of the squares of its elements.

Norm(V1) = sqrt((-1)^2 + 0^2 + 1^2 + 1^2) = sqrt(3) Norm(V2) = sqrt((2/3)^2 + 1^2 + (1/3)^2 + (-2/3)^2) = sqrt(3)

So, the orthonormal basis for the subspace U is:

[ -1/sqrt(3) 0 ] [ 1/sqrt(3) 1/sqrt(3) ]

and

[ 2/(3sqrt(3)) 1/sqrt(3) ] [ 1/(3sqrt(3)) -2/(3*sqrt(3)) ]