Worksheet 18

1. Compute the matrix representation of the linear transformation \(\begin{aligned} T : \mathbb{R}^3 & \to \mathbb{R}^3 \\ \mathbf{v} & \mapsto A \mathbf{v} \end{aligned}\) where \(A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{pmatrix}\) in the basis \(\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) for the source and the target.

2. Suppose we have linear transformations $T,S : V \to W$ and we know that \(T(v_i) = S(v_i) ~\text{for all}~ i\)
   where $v_1,\ldots,v_s$ is a spanning set for $V$. Is $T = S$?

3. Does there exist a change of bases to convert \(\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) to \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} ?\)