Finding out which vectors are linearly dependent

Question

We have following vectors: $(1,2,1,3),(2,-1,3,-1),(3,1,4,2),(1,-3,1,-2)$. I'm trying to find out which of those are linearly dependent and remove them. What I did was to create a matrix: $\begin{pmatrix}
1 & 2 & 3 & 1\\
2 & -1 & 1 & -3\\
1 & 3 & 4 & 1\\
3 & -1 & 2 & -2
\end{pmatrix}$ and by solving the matrix I found out, that the last row will get removed, so they are linearly dependend. However, how do I find out which two vectors are linearly dependent? Thanks

Answer 1

Just reduce it to row echelon form, keeping track of what you do. Start with $$ \begin{pmatrix} 1 & 2 & 3 & 1 &R_1\\ 2 & -1 & 1 & -3 &R_2\\ 1 & 3 & 4 & 1 &R_3\\ 3 & -1 & 2 & -2 &R_4 \end{pmatrix}$$

Then

\begin{pmatrix} 1 & 2 & 3 & 1 &R_1\\ 0 & -5 & -5 & -5 &R_2-2R_1\\ 0 & 1 & 1 & 0 &R_3 -R_1\\ 0 & -7 & -7 & -5 &R_4-3R_1 \end{pmatrix}

and continue in this fashion.