Find a subset that forms a basis for the span of a set Let $p_1 = 1 - 2t - t^2$, $p_2 = t + t^2 + t^3$, $p_3 = 1 - t + t ^3$ and $p_4 = 3 + 4t + t^2 + 4t^3$. 

Let $S$ be the set of these four functions. Find a subset of $S$ that is a basis for the span of $S$. 

So I've started out by taking these functions, making them vectors and putting them into a matrix: 

$$
\begin{bmatrix}
0 &amp; -1 &amp; -2 &amp; 1\\ 
1 &amp; 1 &amp; 1 &amp; 0\\ 
1 &amp; 0 &amp; -1 &amp; 1\\ 
4 &amp; 1 &amp; 4 &amp; 3
\end{bmatrix}
$$

Then I reduced it: 

$$ \sim
\begin{bmatrix}
1 &amp; 0 &amp; 0 &amp; 1\\ 
0 &amp; 1 &amp; 0 &amp; -1\\ 
0 &amp; 0 &amp; 1 &amp; 0\\ 
0 &amp; 0 &amp; 0 &amp; 0
\end{bmatrix}
$$

I guess my question is, is this the correct way to do this? Where do I go from here? What does a zero row mean for a basis?
A better way is to put them into as matrix as columns instead of as rows. The elementary row operations preserve linear dependence relations among the columns, so when you get to reduced form, if, say, columns 1, 2, and 4 are a basis for the column space, then $p_1,p_2,p_4$ are a basis for the original space.
Yes, its the right way, assuming that you reduced your matrix without any mistakes. Now, the zero row mean, that this row are linear dependable on the first three rows, hence your basis for $S$ is $\text{Span}\{p_1,p_2,p_3\}$.