How do I find a basis for span of vectors?

Question

The question asks to find the basis for space spanned by vectors (1, -4, 2, 0), (3, -1, 5, 2), (1, 7, 1, 2), (1, 3, 0, -3).

Answer 1

To do this, simply put the vectors together in a matrix and compute the row-reduced echelon form (rref) of the matrix. Once you have computed the rref, determine the pivot columns — columns that have a leading 1. Then, look at the original matrix. The pivot columns inform you which vectors serve as the basis for the space spanned by the vectors. For instance, if there are pivots in columns 1 and 2, then the first two original vectors serve as a basis for the space.