Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 9), (1, 1, 2), (1, 1, 1)}STEP 1: Find the reduced row-echelon form of the matrix whose rows are the vectors in S. STEP 2: Determine a basis that spans S.

Sure, let's proceed with the steps you've outlined.

STEP 1: We first form a matrix with the vectors in S as rows:

[4 4 9]
[1 1 2]
[1 1 1]

We then reduce this matrix to its row-echelon form using Gaussian elimination.

Subtract the second row from the first, and the third row from the second:

[3 3 7]
[0 0 1]
[1 1 1]

Subtract the third row from the first:

[2 2 6]
[0 0 1]
[1 1 1]

Finally, divide the first row by 2:

[1 1 3]
[0 0 1]
[1 1 1]

STEP 2: The basis for the subspace spanned by S is the set of row vectors in the row-echelon form of the matrix. However, we can see that the first and third rows are linearly dependent (the first row is just the third row multiplied by 3). Therefore, we can remove the third row to get a basis for the subspace:

Basis = {(1, 1, 3), (0, 0, 1)}

This basis spans the same subspace as S, but is a simpler representation because it contains fewer vectors.