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: 
$$
\begin{bmatrix}
0 & -1 & -2 & 1\\ 
1 & 1 & 1 & 0\\ 
1 & 0 & -1 & 1\\ 
4 & 1 & 4 & 3
\end{bmatrix}
$$
Then I reduced it: 
$$ \sim
\begin{bmatrix}
1 & 0 & 0 & 1\\ 
0 & 1 & 0 & -1\\ 
0 & 0 & 1 & 0\\ 
0 & 0 & 0 & 0
\end{bmatrix}
$$
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? 

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 $\text{Span}\{p_1,p_2,p_3\}$. 
