Let $P=F[x]$ be the vector space of polynomials over the field $F$. Determine whether or not $P/M$ is finite dimensional when $M$ is
(i) the subspace $Pn$ of polynomial of degree less or equal $n$;(infinite)
(ii) the subspace $E$ of even polynomials;(infinite)
(iii) the subspace $x^nP$ of all polynomials divisible by $x^n$.(finite)
Let $P$ be as above and $L : P → P$ be given by $L(f(x))=x^2f(x)$.In the examples above, determine whether $L$ induces a map of quotients $L¯:P/M→P/M$ When it does, choose a convenient basis for the quotient space and find a matrix representation of $L¯$.
How to find the basis for the quotient space. for (i), the basis is $(X^{n+1}+M,x^{n+2}+M,...)$ for(ii) $(x+M,x^3+M,...,x^{2n+1}+M...)$ for(iii) $(1+M,x+M,x^2+M...x^{n−1}+M)$ How should I use matrix to represent $L¯$? I've identified that $P/M$ is infinite for i) and ii) and finite for iii). Also $Pn$ and $E$ are not $L$ invariant does this mean that a quotient map won't be induced? I think a quotient map will be induced for iii) but not sure how to find a matrix representation for it. Any help will be appreciated!
If we want to see whether our map $L: P \rightarrow P$ induces an homomorphism $P/M \rightarrow P/M$ we need to check that $L(x+M)= L(x)+M$ is well-defined. I.e. if we pick another representative $x$ the image should not change. Formally
$$ x + M = y + M \Rightarrow L(x) + M = L(y) + M. $$
This is equivalent to
$$ x - y \in M \Rightarrow L(x-y) \in M. $$
For short $L$ maps elements of $M$ to $M$ (i.e. $M$ needs to be invariant under $L$).