The definition of determinat

Main reference: Linear Algebra Done Right by Sheldon Axler.

Definition

For vector space with base field and a fixed basis , we use to denote its all alternating n-froms. It’s a well known fact that this is also a vector space with dimension .

For any endomorphism on , i.e., a matrix, we define a endomorphism of as , the latter applied to a vector in as .

We thus obtain a endomorphism on , since , must be identified with a scalar multiplication , and the constant is unique.

Finally, we define as the unique constant in .

Some quick facts

• The determinat of identity matrix is what you want: by definition.
• Determinat plays well with multiplication: for all morphisms and . Again this is a immediately result by definition.
• Definition plays well with scalar: , by definition.

• Determinat encodes the invertibility of morphisms: For all morphisms , invertible if and only if .

  • Proof: If , we need to prove that is an isomorphism. Thanks to the rank theorem of vector spaes, we only need to prove is injective, which is equivalent to that kernel is trivial. For all non-zero , we can extend it into a basis , choose an alternating n-form , we have , so . Note that for for vectors , if and only if these vectors independant. So, we have that . Since is arbitrary we deduce that injective.

    The other direction is simple, due to determinat plays well with multiplication.