Determinant of Linear Operator is Well Defined

Theorem
Let $V$ be a vector space over a field $K$.

Let $A: V \to V$ be a linear transformation of $V$.

Then the determinant $\det A$ of $A$ is well defined.

Proof
Let $A_\mathcal B$ and $A_\mathcal C$ be the matrices of $A$ relative to $\mathcal B$ and $\mathcal C$ respectively.

Let $\det$ also denote the determinant of a matrix.

We are required to show that $\det A_\mathcal B = \det A_\mathcal C$.

Let $P$ be the change of basis matrix from $\mathcal B$ to $\mathcal C$.

Since $A_\mathcal B$ and $A_\mathcal C$ represent the same linear transformation with respect to different bases, the following diagram commutes:


 * Determinant Independent of Basis.png

where $u \in V$ is some vector, and $\left[{u}\right]_\mathcal B$ indicates the coordinate vector of $u$ with repect to $\mathcal B$, and similarly for $\left[{u}\right]_\mathcal C$.

That is, $P A_\mathcal B = A_\mathcal C P$.

Therefore, because a Change of Basis is Invertible we have:
 * $A_\mathcal B = P^{-1} A_\mathcal C P$

So:

Hence the result.