linear algebra

Just by using the definition of linearly depended and solving the corresponding system of linear equations(with any method you want, usually by Gaussian elimination).

For example to deside if the vectors $v_1=(1,2,3,4),v_2=(2,1,3,4),v_3=(1,2,4,8),v_4=(0,3,4,8)$ are linearly depended you will do:

If $r_1,r_2,r_3,r_4 \in \mathbb{F}$($\mathbb{F}$ being your field) such that $r_1v_1+r_2v_2+r_3v_3+r_4v_4=(0,0,0,0)$ then this is equivalent to $(r_1+2r_2+r_3,2r_1+r_2+2r_3+3r_4,3r_1+3r_2+4r_3+4r_4,4r_1+4r_2+8r_3+8r_4)=(0,0,0,0)$ witch is equivalent to the system of linear equations

\begin{split} r_1&+2r_2+r_3\:\:\:\:\:\;\;\;\;\;\;\;=0\\ 2r_1&+\;\;r_2+2r_3+3r_4=0\\ 3r_1&+3r_2+4r_3+4r_4=0\\ 4r_1&+4r_2+8r_3+8r_4=0 \ . \end{split}

If you solve this system (say, by Gaussian elimination) you will find that it has non zero solution (for example $r_1=1,r_2=-1,r_3=1,r_4=-1$) so

$u_1-u_2+u_3-u_4=0$

and your vectors are linearly depented.