Below is a problem from an online class that I am taking (not for credit or anything so I'm not just asking someone to do my homework; it is just bugging me that I cannot figure it out). I am not sure if there is some kind of gap in my knowledge, but I do not know how I would approach a mapping defined by a matrix whose entries are Laurent polynomials. I feel like I might just be able to solve it as a system but I'm not really sure how proceed.
Find the kernel of the linear transformation $T: \mathbb{C}^3(t)\to \mathbb{C}^3(t)$ defined by the matrix:
$\begin{pmatrix} 0 & 1 & t^{-1}\\ 0 & t & 0\\ 0 & 1 & t^{-1}+t \end{pmatrix}$
EDIT (My seemingly incorrect attempt):
- I am a little confused because the problem said that the mapping acts on the right—doesn't that not work with matrix multiplication? Or am I mixing up the definition of acting on the right? In any case, I did it below with what seems to be the the mapping acting. on the left (?)
Find $\begin{pmatrix}a\\b\\c\end{pmatrix}$ such that
$$\begin{pmatrix} 0 & 1 & t^{-1}\\ 0 & t & 0\\ 0 & 1 & t^{-1}+t \end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}=\begin{pmatrix}1\\0\\0\end{pmatrix}$$ So $$\left(\begin{array}{c} b+\frac{c}{t}\\ b\,t\\ b+c\,\left(t+\frac{1}{t}\right) \end{array}\right) = \begin{pmatrix}1\\0\\0\end{pmatrix}$$ This is impossible, so the kernel is empty?