my professor give an exercise about quotient spaces envolving vector spaces and I have some questions about that.
Consider $f:V \to V$ a linear map and $W \subset V$ a subspace. Take $\phi: V/W \to V/W$, $\phi(v+W) = f(v)+W$.
Questions:
Which conditions is $\phi$ linear map and well-defined?
which is the $\operatorname{Ker}\phi$ and $\operatorname{Im}\phi$??
My observations:
$\phi$ is well-defined: whenever we have $v_1+W = v_2+W$ then $\phi(v_1+W) = \phi(v_2+W)$.
But if $\phi(v_1+W) = \phi(v_2+W) \iff f(v_1) + W = f(v_2) + W \iff f(v_1-v_2) \in W.$ Also, we have $v_1-v_2 \in W.$ Then I conclude that $\phi$ is well-defined if $f(W) \subset W.$ That is it?
For the second question, the kernel is $v + W \in V/W$ such that $f(v+W) = W$ or $f(v+W) \in W$, I don't know which is correct, but it is easy to see that $\operatorname{Ker}(\phi) = \{v+ W: f(v) \in W\}$ ok, but we know this set??? and the image??
Yes, $f(W) \subseteq W$ is a sufficient condition for guarantee that $\phi$ is well-defined. A subspace $W$ of $V$ satisfying that $f(W) \subseteq W$ is called an invariant subspace under $f$, or simply, $f$-invariant. This tool help us to define a new linear operator: $f \upharpoonright W : W \to W$, the restriction of $f$ to the subspace $W$.
Also, your expression for the kernel of $\phi$ is correct, and it could derive like this: $v+W \in \ker(\phi)$ if and only if $f(v)+W = W$, which happens if and only if $f(v) \in W$.
Finally, it it clear that the image of $\phi$ is the set $\operatorname{im}(\phi) = \{w +W\in V/W : w \in \operatorname{im}(f) \}$. Observe that if $f$ is surjective, then $\phi$ as well.