I am writing a first handout on determinants. The intended audience is confident with basic matrix algebra and the basic definitions of vector space theory. I just wondered if someone would comment on my layout and let me know whether they thought it appropriate. Thanks in advance.
$\underline{\text{Handout I}}$
Let $V$ be an $n$-dimensional $F$-vector space.
Definition. Multilinear map.
Definition. Volume form on $V$ i.e. an alternating n-linear map $V^n\to F$.
Proposition. Volume forms are skew-symmetric.
Remark. Converse true unless $\operatorname{char}F = 2$.
Proposition. Let $f$ be a non-zero volume form on $V$. Then $$f(v_1,\ldots,v_n)\neq 0 \Leftrightarrow v_1,\ldots,v_n \text{ is a basis for } V.$$
Definition. Define $\det:\operatorname{Mat}_n(F)\to F$ by $$\det A = \sum_{\sigma \in S_n}\operatorname{sgn}(\sigma)a_{\sigma(1),1}\ldots a_{\sigma(n),n}.$$
Note. We can consider a matrix $A \in \operatorname{Mat}_n(F)$ to be an $n$-tuple of column vectors.
$\hphantom{Note}$ In doing so, we obtain an isomorphism $\operatorname{Mat}_n(F)\cong F^n\oplus\ldots\oplus F^n$.
Proposition. $\det$ is a volume form on $F^n$.
Theorem. Let $f$ be any volume form on $F^n$. Then $$f=f(e_1,\ldots,e_n)\cdot\det.$$ In particular, $\det$ is the unique volume form on $F^n$ whose value at $(e_1,\ldots,e_n)$ is $1$.