A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.

Let $A=[v_1\ \cdots\ v_k]$. Then $Ac=\sum_i c_iv_i$, so the columns are independent iff $\ker A=\{0\}$. This is equivalent to no free variable in $Ac=0$. The matrix may be rectangular; independence requires $k\le m$ for vectors in $\mathbb F^m$.Copy