Let $k$ be a field, let $n \in \mathbb{N}$ with $n \geq 1$, let $V$ be a finite-dimensional $k$-[vector space](/page/Vector%20Space) with $\dim_k V = n$, and let $T: V \to V$ be a $k$-[linear map](/page/Linear%20Map). Define the characteristic polynomial of $T$ by