Advanced Linear Algebra, Lecture 3.3: Alternating multilinear forms

Advanced Linear Algebra, Lecture 3.3: Alternating multilinear forms
A multilinear form is alternating if it is zero whenever two distinct inputs are identical. We show how alternating k-linear forms are skew-symmetric, and the converse holds as long as we are over a field where 1+1≠ 0. After that, we show how if the input vectors to an alternating k-linear form are linearly dependent, the output will be zero. The converse fails -- there are k-linear forms that evaluate to zero on linearly independent sets. However, the converse holds in one important case: when k=n=dim(X), which is a property that we know to hold for determinants. The proof of this actually tells us more -- that any two alternating n-linear forms are scalar multiples of each other. The determinant will end up being the unique alternating n-linear form that is "normalized" to be 1 on the standard unit basis vectors, and this is the topic of the following lecture.
Course webpage: www.math.clemso...

[@MrTroywoo]

21:04 I don't think you can do that for k>2. If you go check Greub's Multilinear Algebra, Ch4, you can see the kernel of the alternator is not symmetric, but something more general.

[@geoffreymilward3293]

Yes, I was struggling to see this. The odd permutations certainly cancel out but why does the sum over even permutations equal the function?

[@lazywarrior]

I have the same question here.

[@fsaldan1]

Shouldn't it be f•π instead if π•f?

[@ProfessorMacauley]

It's fine as is -- I defined π•f to be f(x_1,...,x_n) with the coordinates permuted by π.