please I want to see the solution step by step Let S 1 2 1

please I want to see the solution step by step

Let S = {[-1 -2 1 -3], [-3 -7 3 -9], [-2 -2 1 -4], [1 -7 3 -5], [[3 4 -2 7]}. a. Find a basis for the subspace W spanned by S, and the dimension of W.

Solution

Let A =

-1

-3

-2

1

3

-2

-7

-2

-7

4

1

3

1

3

-2

-3

-9

-4

-5

7

We will reduce A to its RREF as under:

Multiply the 1st row by -1

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

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

Add 3 times the 1st row to the 4th row

Multiply the 2nd row by -1

Multiply the 3rd row by -1

Add -2 times the 3rd row to the 4th row

Add 2 times the 3rd row to the 2nd row

Add -2 times the 3rd row to the 1st row

Add -3 times the 2nd row to the 1st row

Then the RREF of A is

1

0

0

4

-1

0

1

0

1

0

0

0

1

-4

-1

0

0

0

0

0

Apparently, (1,-7, 3,-5)^T=4(-1,-2,1,-3)^T+(-3,-7,3,-9)^T-4(-2,-2,1,-4)^Tand (3,4,-2,7)^T = -(-1,-2,1,-3)^T-(-2,-2,1,-4)^T

Hence a basis for the subspace W spanned by S is {(-1,-2,1,-3)^T, (-3,-7,3,-9)^T, (-2,-2,1,-4)^T}. The dimension of W is 3.

-1	-3	-2	1	3
-2	-7	-2	-7	4
1	3	1	3	-2
-3	-9	-4	-5	7