examples of a special problem in linear transformation A problem asks to find out examples of two distinct linear transformations $T$ and $S$ (say) from vector spaces $V$ to $W$ so that $ker(T)=ker(S)$ and $Im(T)=Im(S)$.
I have given an example from $\mathbb R^2$ to $\mathbb R^2$ where $T(x,y)=(y,x)$ and another is identity transformation.But I want to have examples where domain space and co-domain space are different (maybe of different dimensions). Also my point is to know if there are any transformations existing in  general spaces i.e from $V$ to $W$.
Any help in this regard is welcome.

Here's a generic recipe to create such pairs of linear transformations:


*

*Select a vector space $X$ so that $\dim X\le\min(\dim V, \dim W)$, but otherwise arbitrary.

*Choose an arbitrary surjective linear transformation $A:V\to X$.

*Choose two arbitrary, but different bijective linear transformations $B_1$ and $B_2$ from $X$ to itself.

*Choose an arbitrary injective linear transformation $C:X\to W$.


Then $S = CB_1A$ and $T=CB_2A$.
Basically, $A$ determines the kernel, $C$ determines the image, and $B_1$ and $B_2$ then determine which of the linear transformations with the given kernel and image are chosen.
For example, assume $V=\mathbb R^4$ and $W=\mathbb R^3$. Then you could choose e.g. $X=\mathbb R^2$ and the transformations
$$\begin{align}
A &= \pmatrix{1 & 0 & 0 & 0\\0 & 1 & 0 & 0}\\
B_1 &= \pmatrix{1 & 1\\0 & 1}\\
B_2 &= \pmatrix{2 & 0\\0 & 3}\\
C &= \pmatrix{1 & 0\\0 & 1\\0 & 0}
\end{align}$$
With that choice you'd get
$$\begin{align}
S &= \pmatrix{1 & 1 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 0 & 0}\\
T &= \pmatrix{2 & 0 & 0 & 0\\0 & 3 & 0 & 0\\0 & 0 & 0 & 0}
\end{align}$$
