The map T : M_2x2(R) -> P(2) is not invertible, because the two vector spaces (M_2x2(R) and P(2)) don't have the same dimension. Indeed, if the map had been invertible, then the two vector spaces would have been isomorphic, and then they would have had the same dimension. But M_2x2(R) has dimension 4 and P(2) has dimension 3, whereas they are non-isomorphic.

As for M_2x2(R) and P_3(R) there is a general theorem, which you may know: Two real vector spaces are isomorphic, if and only if they have the same dimension. These two have the same dimension.

Otherwise, map

1 0

0 0

to 1+0x+0x^2+0x^3,

and map

0 1

0 0

to

0+1x+0x^2+0x^3,

and so on. This gives rise to an isomorphism.

For the last question, they are not isomorphic, because their dimensions are unequal.