Reduced row echelon form
Determining if a Vector in a span x is in span y involves making a matrix with the x on one side, y on the other, and checking if any of the x vectors are linear combinations of the y.
Definition:
A system of equations in the variables x1, x2, …, xn. Let [A | b] be a row echelon form of the augmented matrix. xi (i=1, …, n) is a free variable if the corresponding column does not contain a pivot.