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.