Find a basis for the subspace of R3 spanned by S s 5 9 90 1

Find a basis for the subspace of R3 spanned by S. s- (5, 9, 90, (1, 3, 30, (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.

Solution

Let A =

5

1

1

9

3

1

9

3

1

We will reduce A to its RREF as under:

Multiply the 1st row by 1/5

Add -9 times the 1st row to the 2nd row

Add -9 times the 1st row to the 3rd row

Multiply the 2nd row by 5/6

Add -6/5 times the 2nd row to the 3rd row

Add -1/5 times the 2nd row to the 1st row

Then the RREF of A is

1

0

1/3

0

1

-2/3

0

0

0

Apparently, (1,1,1)^T = 1/3(5,9,9)^T -2/3(1,3,3)^T.

Hence, a basis that spans S is { (5,9,9)^T, (1,3,3)^T}.

5	1	1
9	3	1
9	3	1