Can someone please explain why this isn't invertable, I'm having trouble with the definitions.

T: M_2x2(R)->P(2) defined by T a b

C d = a+2bx+(c+d)x^2. (The a b c d is supposed to be a mtrix)

And then could someone explain why the following pairs of vector spaces are isomorphic?

M_2x2(R) and P_3(R)

And why the following pairs are not isomorphic?

V={A$\displaystyle \in$M_2x2(R):tr(A)=0} and R^4