TITLE: Linear Transformations $ \mathbb R^2 \rightarrow \mathbb R^3 $
QUESTION [5 upvotes]: If $ T : \mathbb R^2 \rightarrow \mathbb R^3 $ is a linear transformation such that $ T       \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 12 \\ -2 \end{bmatrix} $ and $ T\begin{bmatrix} 2 \\ -1 \\ \end{bmatrix}  = \begin{bmatrix} 10 \\ -1 \\ 1 \end{bmatrix}  $ then the standard Matrix $A = ?$
This is where I get stuck with linear transformations and don't know how to do this type of operation. Can anyone help me get started ?

REPLY [2 votes]: since $(1,2)$and $(2,-1)$ are independence  vector so they can be a basis for $ R^{2}$.
and if you consider standard basis for $R^{3} \{(1,0,0),(0,1,0),(0,0,1)\}$.
According to Linear Algebra writed by Hofman & Kenzy:
$T_{\text non-stnadard}=[0,10;12,-1;-2,1](2 \times 3$ matrix)
that first column is $T(1,2)$ in standard basis $(T(1,2)=0(1,0,0)+12(0,1,0)+-2(0,0,1)) $
and second column is $T(2,-1)$ at standard basis $(T(2,-1)=10(1,0,0)+-1(0,1,0)+1(0,0,1)). $  
then consider  A=[$\frac{1}{5}$,$\frac{2}{5}$;$\frac{2}{5}$,$\frac{-1}{5}$]
(first column is coordinates of $(1,0)$ in base${(1,2),(2,-1)})$
and second column is coordinate of $(0,1)$ in base $\{(1,2),(2,-1)\}$
$T_{\text non-standard} *A $is matrix $T$ at standard basis .