Thread: [SOLVED] Basis for the subpace V= Span S of R^3?Any ideas?

This is a two part question:

a) Let S= (v1,v2,v3) , where:
v1 = (1,2,3) v2 = (2,1,4) v3 = (1,1,2)
Find a basis for the subspace V= span S of R^3.

I thought I understood how to do this but now I think I'm weong because the second part doesn't apply to my answer. I expressed the vectors as the columns of a matrix and found the basis of the column space to be all 3 vectors which are then the basis for span S. But the second part states:

b) Express each vector not in the basis as a linear combination of the found basis vector.

With my answer all 3 vectors are in the basis so where did I go wrong?

Any help would be appreciated.

Thanks in advance.

Originally Posted by chocaholic

This is a two part question:

a) Let S= (v1,v2,v3) , where:
v1 = (1,2,3) v2 = (2,1,4) v3 = (1,1,2)
Find a basis for the subspace V= span S of R^3.

I thought I understood how to do this but now I think I'm weong because the second part doesn't apply to my answer. I expressed the vectors as the columns of a matrix and found the basis of the column space to be all 3 vectors which are then the basis for span S. But the second part states:

b) Express each vector not in the basis as a linear combination of the found basis vector.

With my answer all 3 vectors are in the basis so where did I go wrong?

Any help would be appreciated.

Thanks in advance.

the set of vectors is linearly independent and is thus a basis for ,which

of course means that in fact and thus you went wrong nowhere.

I suppose part (b)'s answer could be: there's no vector from the original given three ones that is no element of the basis above for so there's no more to be done.

Tonio

Thanks Tonio I thought I was going mad as I kept coming up with the same answer.