Question

1) Decide whether or not the set S of vectors in R3 actually spans R3. If S does not span R find a specific vector int R3 not in the span ()0)0

Answer 1

1). i. Let A be the 3x3 matrix with the vectors in S as columns. Then A =

2	1	-1
1	5	1
-1	4	2

To determine whether the vectors in S span R³, we will reduce A to its RREF as under:

Multiply the 1st row by ½

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

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

Multiply the 2nd row by 2/9

Add -9/2 times the 2nd row to the 3rd row

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

Then the RREF of A is

1	0	-2/3
0	1	1/3
0	0	0

Now it is apparent that the columns of A , and hence the vectors in S are not linearly independent and that (-1,1,2)^T = -(2/3)(2,1,-1)^T + (1/30(1,5,4)^T. Since dim(R³) = 3, the vectors in S do not span R³. Any vector of the form (a,b,c)^T, where c is non-zero,is not in span(S).

ii. Let A be the 3x3 matrix with the vectors in S as columns. Then A =

2	1	0
1	1	1
-1	-1	2

To determine whether the vectors in S span R³, we will reduce A to its RREF as under:

Multiply the 1st row by 1/2

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

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

Multiply the 2nd row by 2

Add 1/2 times the 2nd row to the 3rd row

Multiply the 3rd row by 1/3

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

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

Then the RREF of A is I₃.

It implies that the columns of A , and hence the vectors in S are linearly independent.Further, since dim(R³) = 3, the vectors in S span R³.