Question

Find a basis for the following subspace of R^4. All vectors of the form (a+c, a-b, b+c, -a+b).

Answer 1

There are three parameters here, a, b, and c, so you can start by pulling out the vectors that multiply a,b, and c:

(a+c, a-b, b+c, -a+b) = a(1,1,0,-1) + b(0,-1,1,1) +c(1,0,1,0)

Now check to see if the three vectors are independent by reducing the matrix
that has these vectors as a row

1 1 0 -1 1 1 0 -1 1 1 0 -1
0 -1 1 1 =-> 0 -1 1 1 => 0 -1 1 1
1 0 1 0 0 -1 1 1 0 0 0 0

Since you get zeroes at the bottom, all three vectors aren't independent, which means that there are only two vectors in the basis. Which two of the three vectors you choose is arbitrary, so pick the first to be (1,1,0,-1) and the second to be (0,-1,1,1).

The final check is to notice that the vector you started with can be written as

(a+c, a-b, b+c, -a+b) = (a+c)((1,1,0,-1) + (b+c) (0,-1,1,1)