Advanced Linear Algebra - 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: http://www.math.clemson.edu/~macaule/math8530-online.html