For and , show that there exists a homogeneous system of linear equations whose solution space is W.
Since . Let's say that is a basis of W. Now, construct a matrix A (of size ) such that its rows are elements from the basis of W, stacked together. The row space of A is W, so the row space of its row-echelon form is W too. At this point, I'm stuck! I'm trying to come up with a homogeneous system with the help of A, though there may exist other easier ways of approaching this problem.
Could someone show me the light?
P.S. stands for W is a subspace of .
P.P.S. Isn't this equivalent to saying that W is the null-space of some matrix? Can we go ahead along these lines, and construct a matrix P such that for all ?