Find a basis for the subspace of R3 spanned by S.S = {(1, 2, 6), (−1, 3, 6), (2, 3, 1)}

Sure, let's find a basis for the subspace of R3 spanned by S = {(1, 2, 6), (−1, 3, 6), (2, 3, 1)}.

Step 1: Write the vectors as rows in a matrix We start by writing the vectors in S as rows in a matrix. This gives us the matrix:

1  2  6

-1 3 6 2 3 1

Step 2: Row reduce to echelon form Next, we row reduce this matrix to its echelon form. This involves using elementary row operations to create a matrix where each row has more leading zeros than the row above it.

1  2  6
0  5 12
0  0 -11

Step 3: Identify the basis vectors The non-zero rows of the row reduced matrix correspond to the basis vectors for the subspace spanned by S. So, the basis for the subspace of R3 spanned by S is {(1, 2, 6), (0, 5, 12), (0, 0, -11)}.

Note: The basis of a subspace is not unique. Different row operations could lead to different row reduced matrices, and hence different basis vectors. However, all such bases will have the same number of vectors (the dimension of the subspace), and they will all span the same subspace.