LA: Finding a Basis for a Subspace

[clope023]

Homework Statement

Find a basis for the subspace S of R^4 consisting of all vectors of the form

(a+b, a-b+2c, b, c)^T, where a,b,c are real numbers. What is the dimension of S?

Homework Equations

vectors v1,...,vn from a basis for a vector space iff

i) v1,...,vn are linearly independent
ii) v1,...,vn span V

The Attempt at a Solution

I'm actually not sure how I'd start this problem as most of the basis problems I've been doing have been comparing 2-3 vectors against each other not just one. Would I have to find a vector of scalars in R^4 and find S is both linearly independent and spans R^4 against it? any help is greatly appreciated.

 

[statdad]

Try writing the given vector (the one in your post) as a sum of several, with $$a, b, c$$ as the coefficients of the linear combination. Are the vectors you find linearly independent?

 

[HallsofIvy]

In particular, the zero vector, here (0,0,0,0)^T, is in every subspace. For what values of a, b, c is (a+b, a-b+2c, b, c)^T equal to the vector?