Matrix of transformation $F(x,y,z) = (x,y,z)$

Question

I have proven that $S$ is also a basis. But I am not sure about the second one. Is it just identity matrix $3\times 3$ as we don't change anything?

Answer 1

The representation of the identity transformation is the identity matrix no matter what (ordered) basis is chosen.

Answer 2

The linear transformation $$F(x,y,z) = (x,y,z)$$ is the identity transformation and its matrix with respect to any basis is the identity $3\times 3$ matrix.

$$M=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$$