I've been studying some linear algebra for a test and I've stumbled across this particular issue.
I'm given the transformation matrix $M_{BB'}$, corresponding to some $f \colon \mathbb{R}^n \to \mathbb{R}^m$ with respect to the (ordered) bases $B=(B_1, \dotsc, B_n)$ and $B'=(B'_1, \dotsc, B'_n)$. As I understand, if I have a vector $v$ expressed with respect to the basis $B$, then I can find $f(v)$ by calculating $M_{BB'}v$, obtaining a result expressed in the basis $B'$. Also, if I want to find $f(v)$ for any $v$ expressed in the canonical basis $E$, I can calculate $C_{B'E}M_{BB'}C_{EB}$, where $C_{B'E}$ and $C_{EB}$ are the matrices for the change of basis from $B'$ to $E$ and from $E$ to $B$, respectively, and apply it to the vector $v$, obtaining $f(v)$ with respect to the canonical basis.
As far as finding the image of a particular vector goes, I don't think much else can be said. Nonetheless, when I'm asked to find the image of $f$, given $M_{EE} = C_{B'E}M_{BB'}C_{EB}$, I reason that, since $C_{EB}$ is invertible, instead of computing $C_{B'E}M_{BB'}C_{EB}v$ for a generic $v$, I can assume that one and only one $w = C_{EB}^{-1}v$ exists for each $v$, so it suffices to find the image of $C_{B'E}M_{BB'}v$, i.e., to find the image of $M_{BB'}$, find a basis for this space and transform the basis vectors to the canonical basis.
For the null space of $f$, I'm trying to find all $v$ such that $M_{EE}v = C_{B'E}M_{BB'}C_{EB}v = 0$. And since $C_{B'E}$ is invertible, all I need to solve is: $M_{BB'}C_{EB}v = 0$.
Now, I've tried these "simplified" methods and I get different answers, probably due to a mistake in the calculation, but my questions are the following. Are these methods valid? Is there anything I'm missing? Are these the most efficient ways to compute $Im f$ and $Ker f$? (I'm not really worried about working with large matrices or computational complexity, but I would simply like not to perform unnecessary computations.) Are there other procedures for these goals that I'm not considering? Also, is there any significance as to the "disappearance" of one of the two change of basis matrix in each equation?
Any help or insight into this matter is greatly appreciated.