How do I determine the associated matrix $A_f$ from $\mathbb{R}^2$ to $\mathbb{R}^3$?

Question

I have the following linear mapping defined by $ (, ) = (2 + , − , + 2)$

Determine the associated matrix $_f$

How do you find the associated matrix from $\mathbb{R}^2$ to $\mathbb{R}^3$?

Answer 1

Compute the images under $f$ of the base vectors of $\mathbb{R}^2$,

$$\begin{align}& f(1,0) = (\color{blue}{\star},\color{blue}{*},\color{blue}{\bullet})\\ & f(0,1) = (\color{red}{\star},\color{red}{*},\color{red}{\bullet})\end{align}$$ and place their coordinates (with respect to the base vectors of $\mathbb{R}^3$) in the columns of the matrix: $$A=\begin{pmatrix} \color{blue}{\star} & \color{red}{\star} \\ \color{blue}{*} & \color{red}{*}\\ \color{blue}{\bullet} & \color{red}{\bullet} \end{pmatrix}$$