Editing Determinant (section)

=== Exterior algebra ===
{{See also|Exterior algebra#Linear algebra}}

The determinant of a linear transformation <math>T : V \to V</math> of an <math>n</math>-dimensional vector space <math>V</math> or, more generally a [[free module]] of (finite) [[rank of a module|rank]] <math>n</math> over a commutative ring <math>R</math> can be formulated in a coordinate-free manner by considering the <math>n</math>-th [[exterior algebra|exterior power]] <math>\bigwedge^n V</math> of <math>V</math>.<ref>{{harvnb|Bourbaki|1998|loc=§III.8}}</ref> The map <math>T</math> induces a linear map
:<math>\begin{align}
            \bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\
  v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n.
\end{align}</math>

As <math>\bigwedge^n V</math> is one-dimensional, the map <math>\bigwedge^n T</math> is given by multiplying with some scalar, i.e., an element in <math>R</math>. Some authors such as {{harv|Bourbaki|1998}} use this fact to ''define'' the determinant to be the element in <math>R</math> satisfying the following identity (for all <math>v_i \in V</math>):
:<math>\left(\bigwedge^n T\right)\left(v_1 \wedge \dots \wedge v_n\right) = \det(T) \cdot v_1 \wedge \dots \wedge v_n.</math>

This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on <math>n</math>-tuples of vectors in <math>R^n</math>.
For this reason, the highest non-zero exterior power <math>\bigwedge^n V</math> (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of <math>V</math> and similarly for more involved objects such as [[vector bundle]]s or [[chain complex]]es of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms <math>\bigwedge^k V</math> with <math>k < n</math>.<ref>{{harvnb|Lombardi|Quitté|2015|loc=§5.2}}, {{harvnb|Bourbaki|1998|loc=§III.5}}</ref>