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 $P_n$ 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 $\bar L: P/M → P/M$. When it does, choose a convenient basis for the quotient space and find a matrix representation of $\bar L$.
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 $\bar L$? Thank you so much!