Solution ILT.C32 Chris Black

We have $\lt{T}{\vect{e}_1} = \colvector{1\\2\\1}$ and $\lt{T}{\vect{e}_2} = \colvector{1\\1\\2}$. Putting these into a matrix as columns and row-reducing, we have \begin{align*} \begin{bmatrix} 1 & 1\\ 2 & 1\\ 1 & 2 \end{bmatrix} &\rref \begin{bmatrix} \leading{1} & 0 \\ 0 & \leading{1}\\ 0 & 0 \end{bmatrix} \end{align*} Thus, the set of vectors $\set{\lt{T}{\vect{e}_1},\,\lt{T}{\vect{e}_2}}$ is linearly independent.