Is the set of vectors \begin{equation*} S=\set{ \colvector{2\\-1\\3\\4\\2},\, \colvector{6\\2\\-1\\3\\4},\, \colvector{4\\3\\-4\\-1\\2} } \end{equation*} linearly independent or linearly dependent?
Theorem LIVHS suggests we study the matrix, $A$, whose columns are the vectors in $S$. Specifically, we are interested in the size of the solution set for the homogeneous system $\homosystem{A}$, so we row-reduce $A$. \begin{equation*} A= \begin{bmatrix} 2 & 6 & 4\\ -1 & 2 & 3\\ 3 & -1 & -4\\ 4 & 3 & -1\\ 2 & 4 & 2 \end{bmatrix} \rref \begin{bmatrix} \leading{1} & 0 & -1\\ 0 & \leading{1} & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \end{equation*}
Now, $r=2$, so there are $n-r=3-2=1$ free variables and we see that $\homosystem{A}$ has infinitely many solutions (Theorem HSC, Theorem FVCS). By Theorem LIVHS, the set $S$ is linearly dependent.