Is there any numerical algorithm to find a basis for the subspace, $T^{-1}(\text{im} S)=\{x: T(x)\in \text{im }S\}$ where $T,S:\mathbb{R}^4\to\mathbb{R}^4$ linear map with given matrices $A,E$ as their representation, preferably not invertible.
I thought this way:
Suppose $v_1,\dots,v_k$ be basis for $\text{im} T\cap \text{im} S$ and suppose $x_1,\dots,x_k$ be a basis for $T^{-1}(\text{im} S)$, Then from the definition we need to cho0se $x_1,\dots,x_k$ such that
$T(x_1)=v_1, T(x_2)=v_2,\dots, T(x_k)=v_k$
after choosing $\{x_1,\dots,x_k\}$ then $\{x_1,\dots,x_k\}\cup\ker(T) $
will be a basis for the required set? Am I theoritically right? But I need an algortihm to find a basis. It would be very helpful if someone help me to build up a matlab code for this.
Input: Two $4\times4$ matrices $T,S$.
Output: Basis for $T^{-1}\mathrm{im}S$.
Algorithm: Join the two matrices as $[T,S]$. Find its null-space, that is, the vectors that solve $[T,S]x=0$. These vectors have dimension 8 with the first four components $x'$ and the next four $-y'$. Then $Tx'=Sy'$. Thus extract the $x'$ from the $x$ vectors and make sure they are linearly independent, hence forming a basis.
Note that in your argument, a basis for $T^{-1}\mathrm{im}S$ may have more elements than a basis for $\mathrm{im}T\cap\mathrm{im}S$, since $T$ may map several vectors to one.