Is the set of vectors \begin{equation*} S=\set{ \colvector{2\\-1\\3\\1\\0\\3},\, \colvector{9\\-6\\-2\\3\\2\\1},\, \colvector{1\\1\\1\\0\\0\\1},\, \colvector{-3\\1\\4\\2\\1\\2},\, \colvector{6\\-2\\1\\4\\3\\2} } \end{equation*} linearly independent or linearly dependent?
Theorem LIVRN suggests we place these vectors into a matrix as columns and analyze the row-reduced version of the matrix, \begin{equation*} \begin{bmatrix} 2 & 9 & 1 & -3 & 6\\ -1 & -6 & 1 & 1 & -2\\ 3 & -2 & 1 & 4 & 1\\ 1 & 3 & 0 & 2 & 4\\ 0 & 2 & 0 & 1 & 3\\ 3 & 1 & 1 & 2 & 2 \end{bmatrix} \rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & -1\\ 0 & \leading{1} & 0 & 0 & 1\\ 0 & 0 & \leading{1} & 0 & 2\\ 0 & 0 & 0 & \leading{1} & 1\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{equation*}
Now we need only compute that $r=4<5=n$ to recognize, via Theorem LIVRN that $S$ is a linearly dependent set. Boom!