$T:P_2$ $\rightarrow$ $R^3$ is a linear transformation defined by $$T(a+bx+cx^2) = \left[ \begin{array}{ccc}
2a-b \\
a+b-3c \\
c-a \end{array} \right]$$
This linear transformation is neither one-to-one nor onto, but I don't really get why. I have read the definition of what a one-to-one linear transformation is and what an onto linear transformation is, but I still don't fully grasp the concept. If you could show me why it is neither one-to-one nor onto that would be a big help.
Best Answer
Hint:First note that as $dim(P_2(R))=3=dim(R^3)$ so $T$ is one one iff T is onto{A linear transformation $T:V\to V$ is one-to-one if and only if it is onto} Now show that $T$ is not one to one by showing that ker(T) is nonzero