<s>[INST] How would you determine the transformation matrix? Suppose there exists a linear transformation $T$ where $T: \mathbb{R^3} \to \mathbb{R^5}$ and $T(\textbf{x}) = \text{A} \textbf{x}$. Given
$$ \text{A} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1\\ 2\\ 1 \\3 \end{pmatrix} , 
\quad
\text{A} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} \hspace{6pt}1 \\ \hspace{6pt}2\\ \hspace{6pt}2\\ \hspace{6pt}2 \\-1 \end{pmatrix} ,
\quad
\text{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3\\ 4\\ 3 \\2 \end{pmatrix} ,
$$
how can we then determine the transformation matrix $\text{A}$? What are some "standard methods" to do it? Isn't this the same as finding the change-of-coordinates matrix when making a change of basis?
 [/INST] Let $v_1=(1,1,1)^T, v_2=(1,2,3)^T$ and $v_3=(0,1,1)^T$ then we verify easily that $B=(v_1,v_2,v_3)$ is a basis of $\Bbb R^3$. Let $B_c$ the canonical basis and $P$ the change matrix from $B$ to $B_c$. Now if $\textbf{x}$ is a vector of $\Bbb R^3$ such that $X=(x,y,z)^T$ is its coordinates  in $B_c$ then $PX=(\alpha,\beta,\gamma)^T$ is its coordinates in $B$ and then
$$T(\textbf{x})=A(PX)=\alpha A v_1+\beta A v_2+\gamma A v_3$$
</s>