Find the basis $\mathcal{V}$ of $\mathbb{R}^4$ and $\mathcal{W}$ of $\mathbb{R}^3$. Let $T:\mathbb{R}^4\to\mathbb{R}^3$ be a linear function with the transormation matrix given as:
$$A=\begin{pmatrix}
-3 & 2 & 3 & -3 \\
4 & 0 & -4 & 4 \\
2 & 0 & -2 & 2
\end{pmatrix}$$
relative to the standard basis of $\mathbb{R}^4$ and $\mathbb{R}^3$.
Find the basis $\mathcal{V}$ of $\mathbb{R}^4$ and $\mathcal{W}$ of $\mathbb{R}^3$ with
$$\mathbb{M}_{\mathcal{V},\mathcal{W}}(T)=\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}$$
I have managed to find the lineat function $T$ which is given as:
$$T(x_1, x_2,x_3,x_4)=\begin{pmatrix}
-3x_1+2x_2+3x_3-3x_4 \\
4x_1-4x_3+4x_4 \\
2x_1-2x_3+2x_4
\end{pmatrix}$$
But after that I dont know what to do with the matrix $\mathbb{M}_{\mathcal{V},\mathcal{W}}(T)$. Can someone give me a hint? And I have also looked on the internet and I couldn't find anything helpful.
 A: Let me say something general about this. Let $V$ and $W$ be vector spaces of dimension $\text{dim}(V) = n$ and $\text{dim}(W) = m$. If $f \colon V \rightarrow W$ is a linear map and we denote the rank of $f$ by $r$, then there exist bases $v_1,\dots,v_n \in V$ and $w_1,\dots,w_m \in W$, such that the matrix corresponding to $f$ relative to these bases is of the form
$$
M =\left(
\begin{array}{cc}
E_r & 0 \\ 
0 & 0
\end{array}\right),
$$
where $E_r \in \text{GL}_r(K)$ is the identity matrix. 
How do we find the bases?
Choose a basis $w_1,\dots,w_r \in \text{im}(f)$ and extend it to a basis $w_1,\dots,w_m \in W$. Now choose preimages $v_1,\dots,v_r \in V$ of $w_1,\dots,w_r$. Since the $w_j$ are linear independent, the $v_j$ will also be linear independent and we can extend them to a basis $v_1,\dots,v_n \in V$, where we choose the $n-r$ additional vectors from the kernel of $f$. We can do that since $$n = \text{dim}(V) = \text{rk}(f) + \text{dim}(\text{ker}(f)) = r + (n - r).$$Then by construction the corresponding matrix to $f$ relative to these bases will be of the desired form.
A: I will try to give you some hints. 
If you have $V$  and $ W $ as subspaces of finite vector space of $  V $. 
Let $ (A,B) $  be and $ (A',B')$ be basis   for $V$ and $W$, in that order. 
If you have $T \in L(V,W)$. 
Then: 
$$ T_{A',B'}=S_{B->B'} [T]_{A,B} S_{A'->A} $$ 
So to sum it up, you have to find matrices of basis change to get desired matrix representation of T. You have $T_{A',B'}$ and $T_{A,B}$ , you know your starting basis set is canonical for $R^3$ and $R^4$. 
PS: Also find fundamental subspaces of matrix A! * 
