With three vectors from $\complex{3}$, we can form a square matrix by making these three vectors the columns of a matrix. We do so, and row-reduce to obtain,
\begin{equation*}
\begin{bmatrix}
\leading{1} & 0 & 0\\
0 & \leading{1} & 0\\
0 & 0 & \leading{1}
\end{bmatrix}
\end{equation*}
the $3\times 3$ identity matrix. So by Theorem NME2 the original matrix is nonsingular and its columns are therefore a linearly independent set.