Find bases for the kernel and image of a linear map.

Question

Let $V$ be $\mathbb{C}[x]/(x^3+5x^2+6x+2)$ and let $T:V\rightarrow V$ be defined as $$T(p(x))=(x+1)p(x).$$ Find bases for the kernel and image of $T$.

My attempt: Since $x^3+5x^2+6x+2 = (x+1)(x^2+4x+2)$, it seems clear to me that the kernel ought to be the span of $x^2+4x+2$. However, the image is less clear to me.

I know that we can further decompose the polynomial into $x^3+5x^2+6x+2 = (x+1)(x-r_1)(x-r_2)$ for irrational roots $r_1,r_2$, so is the range just every polynomial without a roots $1,r_1,r_2$?

Any help appreciated!

Answer 1

Indeed, the kernel of $T$ is just the span of $x^2 + 4x + 2$, which is also the ideal generated by $x^2 + 4x + 2$.

In contrast, the image is the ideal generated by $(x+1)$. This ideal has basis $$ \{1(x+1), x(x+1)\} $$ You can tell that we have accounted for the entirety of the image and kernel since their dimensions add up to that of $V$.