Halmos Section 29,30 and 31: Multilinear Forms, Alternating Multilinear Forms, Alternating Multilinear Forms of Maximal Degree
Point of post: In this post we solve the problems given at the end of the sections 29,30 and 31 in Halmos’s book
Problem: Give an example of a vector space and a multilinear form on which is skew-symmetric but not alternating.
Proof: By virtue of a previous theorem it’s clear that that the ground field for have characteristic . So, let’s just consider to be over itself. Then, we can consider
This is evidently bilinear since
since it’s symmetric it follows that is bilinear. Also, it’s clear that it’s skew-symmetric since (recalling that in . It’s not alternating however since .
Problem: Give an example of a non-zero alternating -linear form on an -dimensional space () such that for some linearly independent set of vectors .
Plan of attack: What if we considered a non-zero alternating -form but fixed the last entry. So, to be explicit what if we looked at some non-zero , which we know exists by prior theorem since . Then, we could find some (the superscripts are just meant so you don’t confused these fixed vectors with the variable vectors which are soon to be used) such that . So, what if we considered given by . Indeed this function is multilinear since is multilinear and alternating since is alternating. Moreover, is non-zero since . But, what if we picked a set of vectors which are linearly independent but for which isn‘t? Then since is alternating and is linearly dependent. So with this in mind let’s consider an actual:
clearly then and is non-zero but
even though is a linearly independent set of vectors.
Problem: What is the dimension of all symmetric -linear forms ()? What about the skew-symmetric? What about the alternating ones?
Proof: Personally, and maybe this problem just psyched me out, but I thought this was a tough one to prove. We begin by classifying the skew-symmetric forms.
Claim:Let be an -dimensional space, then the dimension of .
Proof: Fix a basis for . Then, for each with let be the unique -linear form on such that
Then, let where is the “antisymmetrizer function” described in the post on skew-symmetric forms. Note then that if then (note that this no longer evaluates to zero on tuples who have the same elements as but a possibly different order of arrangement) and that . We claim that
To see that suppose that
Note then that for each with we see that
from where linear independence follows. Now, to see that we prove that for any that
to do this it suffices to show that the right hand side equals the left hand side for all elements of . To do this we let be arbitrary. If any of the coordinates of the tuples are equal then both sides of are zero since both sides are skew-symmetric and thus alternating, so assume not. Then, there exists some such that we note then that
Thus, noting that the number of for which is finishes the argument.
Corollary: If the characteristic of the ground field for has characteristic greater than two then
Next we prove the result for symmetric forms, which for some reason I found considerably easier
Claim If is an -dimensional space then
Proof: Fix a basis . It’s clear upon expansion of an arbitrary -linear form though that each such form is really just the linear combination of monomials of the form and that these monomials are linearly independent symmetric forms. The claim follows by noticing that the number of such monomials is . The conclusion follows.
1. Halmos, Paul R. Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print
No comments yet.