Linear transformation and its matrix with respect to unknown bases I am given a linear transformation
$$T:\mathbb{R^3} \rightarrow \mathbb{R^2}$$
$$T((x,y,z)) = (x+y,-y+z)$$
The task if to find a basis in $\mathbb{R^3}$, let's call it $B=\{e_1, e_2, e_3\}$ and $\mathbb{R^2}$, let's call it $B'=\{f_1, f_2\}$ such that $A$ is the matrix of this transformation with respect to the found bases.  
Here is the matrix $A$:
$$A = \begin{bmatrix} 1&0&0\\0&2&0\end{bmatrix}$$
I think I am not sure how to interpret the given matrix $A$.
 A: According to the definition of associated matrix with respect to given bases, you have to find $\{e_1,e_2,e_3\}$ and $\{f_1,f_2\}$ such that
\begin{align}
T(e_1)&=f_1\\
T(e_2)&=2f_2\\
T(e_3)&=0
\end{align}
Note that the problem is undetermined: you can find infinitely many bases with this property.
First find a basis for the kernel of $T$ and you will have $e_3$. Then complete it to a basis for $\mathbb{R}^3$ and define $\{f_1,f_2\}$ according to the specification.
A: You can find the transformation matrix by standard basis of $\mathbb R^3$ and $\mathbb R^2$ called standared basis. $\{ e_1,e_2,e_3\}$ for $\mathbb R^3$ and  $\{ e_1,e_2\}$ for $\mathbb R^2$. now
$T(e_1)=T(1,0,0)=(1,0)=e_1$
$T(e_2)=T(0,1,0)=(1,-1)=e_1-e_2$
$T(e_3)=T(0,0,1)=(0,1)=e_2$
now the transformation matrix with respect to standard basis is:
$$A = \begin{bmatrix} 1&1&0\\0&-1&1\end{bmatrix}$$
Now you could easily find the basis for your question
$T(e_1)=f_1$
$T(e_2)=2f_2$
$T(e_3)=0$
In addition, the nullity of T is spaned by $e_3$.
