Advanced Linear Algebra - Advanced Linear Algebra, Lecture 3.2: Symmetric and skew-symmetric multilinear forms

Advanced Linear Algebra, Lecture 3.2: Symmetric and skew-symmetric multilinear forms A multilinear (or k-linear) form is a scalar function from the direct product of k vector spaces X_1,...,X_k over K, that is linear in each coordinate. In other words, if k-1 entries are fixed, then the resulting function is linear. Note that 1-linear forms are simply scalar functions, and our scalar product notation is an example of a bilinear function. Determinants will end up being a certain type of n-linear form. Most examples that we will see involve X:=X_1=...=X_n. If dim X = n, then the space of k-linear forms is a vector space of dimension n^k. We see why this is true by giving an explicit basis, which generalizes the dual basis of a vector space X. We also see several important classes of k-linear forms: the symmetric and skew-symmetric forms. A multilinear form is symmetric if it is unchanged upon any permutation of its entries. It is skew-symmetric if transposing any two entries flips the sign. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html