$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$
I need to find $e^{T(1+x+x^{2}-x^{3})}$
I don't understand how to do it? what does it mean to calculate exponent of vector? I know the method of doing it for a matrix, finding its Jordan form over the complex field etc.. but what to do here? thank you
The expression $1+x+x^{2}-x^{3}$ is a third-degree polynomial in $x$. The expression $T(1+x+x^{2}-x^{3})$ is also a third-degree polynomial in $x$. You can evaluate such a polynomial at any given value of $x$. So $e^{T(1+x+x^{2}-x^{3})}$ is a function of $x$ described as follows:
$$e^{T(1+x+x^{2}-x^{3})} = e^y$$
where $y = T(1+x+x^{2}-x^{3})$. That is, $y$ is the number obtained by evaluating the polynomial $T(1+x+x^{2}-x^{3})$ at a given value of $x$.