Question

In the real vector space R ³, the vectors u1 =(1,0,0) and u2=(1,2,0) are known to lie in the span W of the vectors w1 =(3,4,2), w2=(0,1,1), w3=(2,1,1) and w4=(1,0,2). Find wi, wj ?{w1,w2,w3,w4} such that W = span({u1,u2,wk,wl}) where {1,2,3,4}= {i,j,k,l}.

Answer 1

Let A = [u₁,u₂, w₁,w₂,w₃,w₄] =

1	1	3	0	2	1
0	2	4	1	1	0
0	0	2	1	1	2

The matrix A can be reduced to its RREF as under:
multiply the 2nd row by ½

Multiply the 3rd row by ½

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

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

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

Then the RREF of A is

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

Apparently, the vectors u₁,u₂, w₁ are linearly independent and the vectors w₂,w₃,w₄ are linear combinations of u₁,u₂, w₁. Also, span { u₁,u₂, w₁}= R³.

Further, let M = [w₁,w₂,w₃,w₄] =

3	0	2	1
4	1	1	0
2	1	1	2

The RREF of M is

1	0	0	-1
0	1	0	2
0	0	1	2

Thus, w₄ is a linear combination of w₁,w₂,w₃ so that span { w₁,w₂,w₃,w₄} = span { w₁,w₂,w₃}= R³. Hence, span {u₁,u₂, w₁}= W = span { w₁,w₂,w₃,w₄} = R³.