Basis for subspace from a set of vectors.

Hello everyone,

I'm studying for my exam in a few days, i stumbled on a question. I researched really hard in the reference book i use (Anton's Elementary Linear Algebra), and searched on the internet on general Basis and Vector Spaces. (I've been researching for 3 hours)

The problem says ...

Find a basis for the subspace of R^4 spanned by the given vectors:

a) (1,1,-4,-3), (2,0,2,-2), (2,-1,3,2)

b) (1,1,0,0), (0,0,1,1), (-2,0,2,2), (0,-3,0,3)

I have a solution given by the doctor, but i just don't get it really. So i opted to research myself.

I know how to prove if a set of vectors are a basis for a vector space, and how to get a basis for the solution of a system of equations, which is what i thought about doing here.

For example, i thought about arranging those vectors in a matrix, row after row, then multiplying it by a 1x4 matrix of any arbitrary variables, kind of like getting the solution for a system of equations, but you're given the variables in that case. Then equate it by another arbitrary vector? Then get the solution in Linear Combination form, and the resulting vectors should be the basis, right?

The doctor arranged that said matrix, reduced it, took the resulting rows as the basis, does that even make any sense?

Anyway, i proceeded with my idea, it made the solution very complex to simplify because of the lone matrix i added. Just for the fun of it, i removed it, and put a zero matrix, the result was a basis of a 1-dimensional space ...

a

b

c

d

vector

Any help would be appreciated please, cause i feel i'm not solving methodically at all. Thank you.