<s>[INST] Finding out which vectors are linearly dependent 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
 [/INST] 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. 
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