Finding transformation matrix given the image of 3 vectors.

[obsidian flume]

Let $g:\mathbb R^3\to\mathbb R^4$ be a linear transformation such that,
$$g\left(\begin{bmatrix}1\2\1\end{bmatrix}\right)=\begin{bmatrix}1\1\2\3\end{bmatrix},g\left(\begin{bmatrix}2\1\1\end{bmatrix}\right)=\begin{bmatrix}1\1\2\3\end{bmatrix},g\left(\begin{bmatrix}1\1\1\end{bmatrix}\right)=\begin{bmatrix}1\1\1\4\end{bmatrix}$$Find the matrix $B$ such that $B \vec x=g(\vec x) , \forall\vec x\in\mathbb R^3$.

[hardy blade]

trigonometria

[obsidian flume]

My intuition tells me $$BV=Y\iff B=YV^{-1}.$$ Where $$V:=\begin{bmatrix}1&2&1\2&1&1\1&1&1\end{bmatrix}\text{and } Y:=\begin{bmatrix}1&1&1\1&1&1\2&2&1\3&3&4\end{bmatrix}$$
I'm not going to bother you with further calculations.

[hardy blade]

trigonometria