TITLE: Why is this not an isomorphism?
QUESTION [7 upvotes]: Let $T(f(t))$=
$\begin{bmatrix}
f(0) & f(1)\\ 
f(2) & f(3)
\end{bmatrix}$ from $P_2$ to $\mathbb{R}^{2\times 2}$. To show that it is not an isomorphism, I need to show that either kernel of the transformation is not equal to the zero element only, or that the image is not the whole target space. I am struggling in showing that either of these is false, dealing with polynomials in transformations is very counter-intuitive. Thanks for help!

REPLY [9 votes]: A polynomial of degree at most 2 is determined by its values at three points. So if $f(0)=f(1)=f(2)=0$, then $f(3)=0$ also. Thus $T$ is not surjective.