Show that two sets of vectors span the same subspace

[csc2iffy]

Homework Statement

Show that the two sets of vectors
{A=(1,1,0), B=(0,0,1)}
and
{C=(1,1,1), D=(-1,-1,1)}
span the same subspace of R3.

Homework Equations

{A=(1,1,0), B=(0,0,1)}
{C=(1,1,1), D=(-1,-1,1)}

The Attempt at a Solution

aA+bB=(a,a,0)+(0,0,b)=(a,a,b)
aC+bD=(a,a,a)+(-b,-b,b)=(a-b,a-b,a+b)
I am confused because I thought the answers would turn out to be equal.. Is this not the way to do it?

 

[radou]

Well, take (a-b, a-b, a+b). Now, this equals a(1, 1, 1) + b(-1, -1, 1). Of what form are the vectors (1, 1, 1) and (-1, -1, 1)? Do they look familiar?

 

[csc2iffy]

its aC+bD... i don't understand how this is related to aA+bB??

 

[radou]

The two vectors I mentioned are in your space "(a, a, b)". Hence, their linear combination is in that space, too.

 

[csc2iffy]

Oh! Thank you! now i see it lol