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?


Isn't this the same as finding the change-of-coordinates matrix when
  making a change of basis?

Totally.
Let $P = \begin{pmatrix} 1 & 1& 0 \\ 1 &2 & 1 \\ 1 & 3& 1\end{pmatrix}$
It's easy to see that $P\in GL_3(\mathbb R)$.
Your equalities can be rewrite: 
$$AP= \begin{pmatrix} 1&1&2 \\ 1&2&3\\ 2&2&4\\ 1&2&3 \\3 &-1&2\end{pmatrix}$$
Finally:
$$ A =  \begin{pmatrix} 1&1&2 \\ 1&2&3\\ 2&2&4\\ 1&2&3 \\3 &-1&2\end{pmatrix} P^{-1}$$
