Basics of Algebra and Analysis for Computer Science by Gallier J.
By Gallier J.
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E → F is called alternating, if f (x1 , . . , xn ) = 0 whenever xi = xi+1 , for some i, 1 ≤ i ≤ n − 1 (in other words, when two adjacent arguments are equal). It does not harm to agree that when n = 1, a linear map is considered to be both symmetric and alternating, and we will do so. When n = 2, a 2-linear map f : E1 × E2 → F is called a bilinear map. We have already seen several examples of bilinear maps. Multiplication ·: K × K → K is a bilinear map, treating K as a vector space over itself.
Xi , . ) = f (. . , xi + λxj , . ), for any λ ∈ K, and where i = j. Proof . (1) By multilinearity applied twice, we have f (. . , xi + xi+1 , xi + xi+1 , . ) = f (. . , xi , xi , . ) + f (. . , xi , xi+1 , . ) +f (. . , xi+1 , xi , . ) + f (. . , xi+1 , xi+1 , . ), and since f is alternating, this yields 0 = f (. . , xi , xi+1 , . ) + f (. . , xi+1 , xi , . ), that is, f (. . , xi , xi+1 , . ) = −f (. . , xi+1 , xi , . ). (2) If xi = xj and i and j are not adjacent, we can interchange xi and xi+1 , and then xi and xi+2 , etc, until xi and xj become adjacent.
En → F is a multilinear map (or an n-linear map) if it is linear in each argument, holding the others fixed. More explicitly, for every i, 1 ≤ i ≤ n, for all x1 ∈ E1 . , xi−1 ∈ Ei−1 , xi+1 ∈ Ei+1 , . , xn ∈ En , for all x, y ∈ Ei , for all λ ∈ K, f (x1 , . . , xi−1 , x + y, xi+1 , . . , xn ) = f (x1 , . . , xi−1 , x, xi+1 , . . , xn ) + f (x1 , . . , xi−1 , y, xi+1 , . . , xn ), f (x1 , . . , xi−1 , λx, xi+1 , . . , xn ) = λf (x1 , . . , xi−1 , x, xi+1 , . . , xn ). When F = K, we call f an n-linear form (or multilinear form).