How to express a vector as a linear combination of others? I have 3 vectors, $(0,3,1,-1), (6,0,5,1), (4,-7,1,3)$, and using Gaussian elimination I found that they are linearly dependent. The next question is to express each vector as a linear combination of the other two. Different resources say just to use Gaussian elimination, but I just end up with a matrix in RREF. How can I find different vectors as a linear combination of others?
 A: Let's look at Gaussian elimination:
\begin{align}
\begin{bmatrix}
0 & 6 & 4 \\
3 & 0 & -7 \\
1 & 5 & 1 \\
-1 & 1 & 3
\end{bmatrix}
\xrightarrow{\text{swap row 1 and 3}}{}&
\begin{bmatrix}
1 & 5 & 1 \\
3 & 0 & -7 \\
0 & 6 & 4 \\
-1 & 1 & 3
\end{bmatrix}\\
\xrightarrow{R_2-3R_1}{}&
\begin{bmatrix}
1 & 5 & 1 \\
0 & -15 & -10 \\
0 & 6 & 4 \\
-1 & 1 & 3
\end{bmatrix}\\
\xrightarrow{R_4+R_1}{}&
\begin{bmatrix}
1 & 5 & 1 \\
0 & -15 & -10 \\
0 & 6 & 4 \\
0 & 6 & 4
\end{bmatrix}\\
\xrightarrow{-\frac{1}{15}R_2}{}&
\begin{bmatrix}
1 & 5 & 1 \\
0 & 1 & 2/3 \\
0 & 6 & 4 \\
0 & 6 & 4
\end{bmatrix}\\
\xrightarrow{R_3-6R_2}{}&
\begin{bmatrix}
1 & 5 & 1 \\
0 & 1 & 2/3 \\
0 & 0 & 0 \\
0 & 6 & 4
\end{bmatrix}\\
\xrightarrow{R_4-6R_2}{}&
\begin{bmatrix}
1 & 5 & 1 \\
0 & 1 & 2/3 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}\\
\xrightarrow{R_1-5R_2}{}&
\begin{bmatrix}
1 & 0 & -7/3 \\
0 & 1 & 2/3 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}\\
\end{align}
If $v_1$, $v_2$ and $v_3$ are your vectors, this says that
$$
v_3=-\frac{7}{3}v_1+\frac{2}{3}v_2
$$
because elementary row operations don't change linear relations between the columns.
A: Since (0,3,1,-1) is a linear combination of (6,0,5,1) and (4,-7,1,3), we can write 
$(0,3,1,-1) = a(6,0,5,1) + b(4,-7,1,3)$. This gives us 4 relations to solve for a and b. We can proceed similarly for the other two vectors.
