Find a basis for W = {(x,y,z,w): x − 2 y + w = 0, 2 x − z + w = 0}.

Question

From what I know, to find a basis for the span of a set of vectors, write the vectors as rows of a matrix and then row reduce the matrix.

However,

[1,0,-1/2,1/2,0], and [0,1,-1/4,-1/4,0] isn't correct.

Answer 1

There are 4 variables and 2 equations. The basis is going to consist of 2 vectors which are not unique of course. From the second equation we get that w=z-2x and from the first we get that w=2y-x. Hence, 2y-x=z-2x. Solving for z yields z=2y+x. Now that we have z,w in terms of x and y the basis vectors are (x,y,z,w)=(x,y,2y+x,2y-x)=x(1,0,1,-1)+y(0,1,2,2) for x,y free variables.