Advanced Linear Algebra, Lecture 3.4: The determinant of a linear map

Advanced Linear Algebra, Lecture 3.4: The determinant of a linear map

In this lecture, we show that there is always a nonzero alternating n-linear form, and along with our previous result that any two such forms are linearly independent, we conclude that the subspace of alternating n-linear forms is 1-dimensional. Given such a form f, every linear map T from a vector space X to itself defines a new alternating n-linear form, defined by first applying T to each entry. This is a linear map on the 1-dimensional subspace of such forms, and so it is simply a scalar function f→λf. The constant λ is the determinant of T. We restate this result in the language of universal properties, and then finish by proving a few basic properties about the determinant in this basis-free language.

[fsaldan]

On 18' I don't see that l(x5)=0. But you don't need it. If x4=x5 then the two remaining terms equal f(x1, x2, x3, z)l(z) except for the sign, with z=x4=x5, and are negatives of each other.