<s>[INST] What is the fastest way of calculating whether vectors are linearly independent? 
Is the following family $(u_1, u_2, u_3)$ linearly independent?
$$ u_1 := \begin{pmatrix} 1 \\ 3 \\ 5 \\ -1 \end{pmatrix} ,
 u_2 := \begin{pmatrix} 1 \\ -1 \\ -3 \\ 3 \end{pmatrix},
 u_3 := \begin{pmatrix} 3 \\ 2 \\ 1 \\ 4 \end{pmatrix}$$

So, do I have to form a matrix $U$ with columns $u_1$, $u_2$ and $u_3$ and use gaussian elimination to see whether in every column is a pivot or is there a faster way? The problem is meant to be solved fast, that is why I am asking. I have the feeling that I am missing some important information.
 [/INST] With Gaussian elimination, it is not very long to prove the rank is $2$:
\begin{align}
\begin{bmatrix}
1&1&3\\ 3&-1&2\\  5&-3&1\\-1&3&4
\end{bmatrix} \rightsquigarrow 
\begin{bmatrix}
 1&3&1\\-1&2&3\\-3&1&5\\3&4&-1
\end{bmatrix} \rightsquigarrow 
\begin{bmatrix}
 1&3&1\\ 0& 5 & 4\\-3&1&5\\ 0 & 5& 4
\end{bmatrix}  \rightsquigarrow 
\begin{bmatrix}
 1&3&1\\ 0& 5 & 4\\ 0&10&8\\ 0 & 0& 0
\end{bmatrix} \rightsquigarrow 
\begin{bmatrix}
 1&3&1\\ 0& 5 & 4\\ 0&0&0 \\ 0 & 0& 0
\end{bmatrix}
\end{align}
</s>