<s>[INST] Isomorphism of a linear map of polynomials Let $V$ be the space of polynomials in one variable of degree $\leq3$.
Let $T$ be a map assigning to $p(x)$ the polynomial $ap(x)+(bx+c)p'(x)+(dx^2+ex+f)p''(x)$, for some real parameters $a, b, . . . , f.$
What are the set of parameters when $T$ is not an isomorphism?
I think I need to show that Ker$T$ contains only the zero vector, but I'm not sure whether this is correct or if it is, how to do this.
In trying to find the kernel I have found $T(a_0+a_1x+a_2x^2+a_3x^3)$ and simplified this giving me $$T(a_0+a_1x+a_2x^2+a_3x^3)=aa_0+ca_1+2fa_2+x(aa_1+a_1b+2ca_2+2ea_2+6fa_3)+x^2(aa_2+2a_2b+3ca_3+2da_2+6ea_3)+x^3(aa_3+3ba_3+6da_3)$$
I'm not really sure where to go now with it though. 
 [/INST] Let $T$ be $T(p(x))=a \cdot p(x)+(bx+c)\cdot p'(x)+(dx^{2}+ex+f)\cdot p''(x)$
Then, $T$ is a linear map as $T(k\cdot p(x))=k\cdot T(p(x))$ and $T(p(x)+q(x))=T(p(x))+T(q(x))$. So, $T$ can be expressed in the form of matrix(with respect to the basis $1, x, x^{2}, x^{3}$)
Thus, if we calculate $T$ with the fact that differentiation operator is expressed as
$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3\\ 0 & 0 & 0& 0 \end{pmatrix}$
We can say that
$T= a\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0& 1 \end{pmatrix}+b\begin{pmatrix} 0& 0&0&0&\\0&1&0&0\\0&0&2&0\\0&0&0&3\end{pmatrix}+c\begin{pmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\end{pmatrix}+d\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&2&0\\0&0&0&6\end{pmatrix}+e\begin{pmatrix}0&0&0&0\\0&0&2&0\\0&0&0&6\\0&0&0&0\end{pmatrix}+f\begin{pmatrix}0&0&2&0\\0&0&0&6\\0&0&0&0\\0&0&0&0\end{pmatrix}=\begin{pmatrix}a&c&2f&0\\0&a+b&2c+2e&6f\\0&0&a+2b+2d&3c+6e\\0&0&0&a+3b+6d\end{pmatrix}$
Thus, $\det(T)=a(a+b)(a+2b+2d)(a+3b+3d)$ should not be zero if and only if $T$ is an isomorphism. 
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