Solution ILT.C31 Chris Black

We have \begin{align*} \lt{T}{\vect{e}_1} &= \colvector{2\\0\\1} & \lt{T}{\vect{e}_2} &= \colvector{1\\2\\0} & \lt{T}{\vect{e}_3} &= \colvector{0\\1\\2} \end{align*} Let us put these vectors into a matrix and row reduce to test their linear independence. \begin{align*} \begin{bmatrix} 2 & 1 & 0\\ 0 & 2 & 1\\ 1 & 0 & 2 \end{bmatrix} &\rref \begin{bmatrix} \leading{1} & 0 & 0\\ 0 & \leading{1} & 0\\ 0 & 0 & \leading{1} \end{bmatrix} \end{align*} so the set of vectors $\set{\lt{T}{\vect{e}_1},\, \lt{T}{\vect{e}_1},\,\lt{T}{\vect{e}_1}}$ is linearly independent.