Change of Basis between different dimension

Question

Changing basis from 3 dimension to 2 dimension is possible?

I’ve been trying to solve this but I couldn’t get this. Please help me!!

Answer 1

$$T\left( \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \right) = \left( \begin{array}{c} 1 \\ 0 \end{array} \right),$$ and $$T\left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right) = -\left( \begin{array}{c} 1 \\ 0 \end{array} \right).$$

So the image of the first basis element of $B$ is just the basis element of $C$. Thus the first column of the $1 \times 2$ matrix is $(1)$. Similarly, the image of the second basis element of $B$ is minus the basis element of $C$, so the second column of the $1 \times 2$ matrix representing $T$ with respect to $B$ and $C$ is $(-1)$. Hence $T$ is represented by $(1 -1)$ with respect to the bases $B$ and $C$. This is the answer to part $1$. I hope this helps. Good luck preparing for your exam.