Exterior Algebra: determinant and induced operator

Question

Let $A: V \to V$ linear operator where $V$ is $n$-dimensional vector space. Consider $\wedge^{k}A: \wedge^{k}V \to \wedge^{k}V$ given by $u_{1}\wedge ... \wedge u_{k} \mapsto A(u_{1})\wedge ... \wedge A(u_{k})$.

When $k=n$, we know that $A(u_{1})\wedge ... \wedge A(u_{n}) = det(A) (u_{1}\wedge ... \wedge u_{n})$.

But what can we say when $k<n$?

(In wikipedia, they say that " Minors of a matrix can also be cast in this setting, by considering lower alternating forms $\wedge^{k}V$ with $k < n$.", but they don't give any reference.)

I'd like some reference to study it. Thanks

Answer 1

See my answer here.

Two good references are:

Bourbaki, Algebra I: Chapters 1-3, Proposition 10, page 529.

Birkhoff and MacLane, Algebra, pages 563-564.