Exterior Algebra - Functoriality

Functoriality

Suppose that V and W are a pair of vector spaces and f : V → W is a linear transformation. Then, by the universal construction, there exists a unique homomorphism of graded algebras

such that

In particular, Λ(f) preserves homogeneous degree. The k-graded components of Λ(f) are given on decomposable elements by

Let

The components of the transformation Λ(k) relative to a basis of V and W is the matrix of k × k minors of f. In particular, if V = W and V is of finite dimension n, then Λn(f) is a mapping of a one-dimensional vector space Λn to itself, and is therefore given by a scalar: the determinant of f.