## Bilinear Forms

**Point of post:** In this post I give a quick overview of the very basic concepts in the topic of “bilinear forms”. The main things covered being what a bilinear form on the product of two vector spaces is, and finding a basis for it. Incidental little facts may pop up.

*Remark:* If are vector spaces, we follow Roman’s (in the author’s opinion) better notation for the external direct sum and reserve for the internal direct sum. Also, we shall use the notation to denote “ is a linear subspace of “. Lastly, for we define

**Motivation**

We know that given a vector space over a field it is fruitful to study the dual space of , denoted as always or , which is the set of all linear mappings . We have also seen that it is interesting to study given vector spaces over a field . It makes sense then to ask questions about maps . This kind of mapping has arisen before in our studies, for example the map

Even more common, is the dot product on usual real coordinate space

Maybe less obvious, but another very important example is the determinant of complex matrices when thought of as being functions of the column vectors, namely:

**Bilinear Forms**

All of these are certainly very important, and commonplace functions the working mathematician encounters on a day-to-day basis. So the question becomes, how do these maps relate, are they specific examples of some larger definable class of maps on vector spaces? Intuition first may tell us that they are linear functionals on the external direct product , but a quick check botches this idea, namely:

But, a little bit of digging reveals that while they aren’t linear in both entries simultaneously they are linear in each entry individually. For example, one can easily check that

More generally we call a mapping

(where is any unspecified vector space over ) *bilinear* if

And

If it happens that these are mappings into the ground field , symbolically:

we call a *bilinear form*. Somethings become immediately apparent from this definition

*Remark: *From here on out, unless specified otherwise, we will have that are vector spaces over a field and is a bilinear form.

**Theorem:** * for any and .*

**Proof:** This follows immediately since , and so . Similarly, and so .

So, we can consider

Consider though that if and then if we define

and

we can easily check that . Thus, we may define and , and so noticing that

is bilinear we can easily conclude that (where this latter space is the set of all mappings of the form ).

**Basis and Dimension**

With this insight, the first question is, if and what is ? But, finding dimension is synonymous (for most finite-dimensional cases) with finding a basis for the space. Thus, our real goal is to ascertain whether given a basis for and a basis for how we can create a related basis for . The first step in this process is the following theorem:

**Theorem: ***For any there exits a unique such that .*

**Proof: **We first notice that if and that and . Thus:

So, define

Evidently then and so the existence is clear, and uniqueness follows by considering if were another such bilinear form then using we may see that

and so .

With this we are now able to produce an explicit basis for given a basis for and one for .

**Theorem:** *Let be a basis for and a basis for . Then,*

*is a basis where*

**Proof:** The fact that such exists is a consequence of the prior theorem. Suppose then that

Then, evaluating at gives

Thus, ranging over and over we find that

and thus is linearly independent. To see that we let be arbitrary. We claim that

To see this we note that if and that

Thus, is linearly independent and , thus is a basis for as required.

From this we may gather the following rather aesthetically appealing corollary:

**Corollary:**

**References:**

1. Halmos, Paul R. “Bilinear Forms.” *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print.

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