Textbooks in which determinant is defined as an alternating multilinear map

Question

I'm interested in this abstract definition of determinant, i.e. determinant is defined as an alternating multilinear map. Could you please suggest me some Linear Algebra textbooks that define determinant in such way.

Thank you so much for your references!

Answer 1

Linear Algebra by Hoffman and Kunze does determinant theory in the way you mentioned. Further it treats matrices not just over fields but over arbitrary commutative rings with unity.

Answer 2

Chapter 3 of Treil's "Linear Algebra Done Wrong" puts a few more desirable conditions of the determinant, and then shows that their conditions are sufficient to serve as a definition. (For instance, it demands that the determinant of the unit diagonal matrix is 1.)

Answer 3

That's how it's done in my favorite Linear Algebra textbook: Katsumi Nomizu's Fundamentals of Linear Algebra. See its sixth chapter.

Answer 4

Loomis and Sternberg's Advanced Calculus has a nice small chapter on multilinear algebra (Chapter 7), which includes the definition for determinant of matrices as the unique alternating multilinear functional with $D(e_1, \dots, e_n) = 1$ (the uniqueness coming from the fact that this subspace has dimension $1$). Then, it also defines the determinant of a linear operator $T:V \to V$ without ever referring to a basis (using of course the fact that top-degree alternating tensors form a one-dimensional vector space), then finally relates the two approaches.

Although this is not a linear-algebra book, it treats the subject very clearly. One thing you may or may not like is that they only consider vector spaces over $\Bbb{R}$ (because it's a book on multivariable calculus/basic differential geometry so that's all they need). But you can easily modify several arguments to hold over general fields (or else, just refer to a more specialized linear algebra text as suggested by others).