Alternating Forms of Maximal Degree
Point of post: In this post I discuss the topic of alternating forms of maximal degree as is discussed in Halmos’s section 31. I also give proof of the last theorem discussed in section 30.
In this post we give two quick observations about alternating forms -forms on an -dimensional space. This will show, in essence that the space of alternating -linear forms has dimension . So, the alternating forms make up a relatively small slice, loosely speaking, of the entirety of . The title is named as it is because we call a -linear alternating form on an -dimensional space an alternating form of maximal degree. The reason is that, as we shall see, if the only -linear alternating form is the zero form.
Theorem: Let and be a -dimensional -space. Then, if is the set of all -linear, skew-symmetric forms on then is a subspace of .
Proof: Evidently is non-empty since . So, now suppose that and . Note then that if and then
and since and were arbitrary it follows that is alternating and thus in.
We now claim that . To do this, we’ll show that any two alternating forms of maximal degree are linearly dependent.
Theorem: Let be a -dimensional -space with basis and . Then, and are linearly dependent.
Proof: Note that since any field is a -dimensional vector space over itself it follows that the two scalars and are linearly dependent and so there exists a non-zero scalar such that . Note then by an earlier theorem we have that for any it’s true that
and since were arbitrary it follows that as desired.
We now prove that in fact, if then by showing the existence of a non-zero alternating -form. We follow the proof given in Halmos closely:
Theorem: Let be a -dimensional -space and . Then, is non-trivial.
Proof: We proceed by induction. For this is true since evidently . Now, suppose that for some and thus has some non-zero . Now since we may find such that . So, now since we may find some and so evidently we may find some such that and (for example, we may extend to some basis (since they must be l.i. since the value of an alternating multilinear form on a dependent set is zero) and so we may define to be the unique linear functional such that and ).
We then define by
We now claim that is non-zero and alternating. The first fact is clear since
We suppose next that are vectors and (distinct) and . Note that if occur simultaneously in the slots of in the expression for then that term is zero.
From here we break into two cases, namely and . If the first is true then all the terms of vanish except
and since the transposition does not change the value of and thus the above vanishes. If then the number of terms in which are non-vanishing is two, and since one of these two may be obtained by applying to the other it follows that it vanishes as well. Thus, is alternating and thus the induction is complete.
Corollary: If is a -dimensional -space, then
Proof: Since we proved any two -forms must be linearly dependent it follows that and since, as was just proven, is not trivial it follows that .
References: Halmos, Paul R. Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print