Originally Posted by
adam_leeds Consider the vectors:
[-1,1,3]
[2,-1,0]
[1,1,5]
[-5,1,1]
Find a subset of the vectors that forms a basis for the space spanned by the vectors.
Method: make the vectors into the rows of a matrix.$\displaystyle \begin{bmatrix}-1&1&3\\ 2&-1&0\\ 1&1&5\\ -5&1&1\end{bmatrix}$
Carry out a Gaussian row-reduction on the matrix. Ignore any rows that end up as consisting of all zeros. The remaining (nonzero) rows tell you which of the original vectors form a basis for the space that they span. Namely, you take the vectors that were originally in those nonzero rows.