The question shows a linear transformation and asks to show that it is isomorphic. I understand the one-to-one part, but don't understand the onto part. The solution manual explains it this way :
What does "the image of this member of the domain..." mean? I can't understand it and don't get how it proves the function is onto.
I apologize if the question is too simple or if I'm asking a dumb question.
It means that$$f\bigl((s+t)+tx\bigr)=\begin{pmatrix}s\\t\end{pmatrix}.$$This proves that every element of $\mathbb R^2$ can be written as $f\bigl(P(x)\bigr)$, for some $P(x)\in\mathcal P_1$; in other words, $f$ is surjective.