Is a multilinear form/mapping a product of some type on vectors?

Question

Added: are all types of mappings for vector spaces with "product" in their names always multilinear mappings between some vector spaces? Are there many counterexamples?

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$F$ is a field. Any bilinear form on $F^n$ can be expressed as $$ B(\textbf{x},\textbf{y}) = \textbf{x}^\mathrm T A\textbf{y} = \sum_{i,j=1}^n a_{ij} x_i y_j $$ where $A$ is an n × n matrix.

I was wondering if $\textbf{x}^\mathrm T A\textbf{y}$ is a single product of some type between two vectors in $F^n$? What type is it?

More generally, is a multilinear form on $F^n$ a single product of some type on multiple vectors in $F^n$?

Is a multilinear form defined on the product space of a vector space a single product of some type on multiple vectors in the vector space?

Is a multilinear form defined from the product space of a vector space to another vector space a single product of some type on multiple vectors in the first vector space?

Thanks and regards!

Answer 1

Well, the overriding question here is what you mean by "product". Generally, a "product" on some structure $V$ (in this case, a vector space) is a binary operation of a sort; i.e. a mapping $$ V \times V \rightarrow V, $$ possibly satisfying some conditions (i.e. associativity, commutativity). As you have things set up, a bilinear form isn't going to be a "product" in this sense, because the codomain isn't right: a bilinear form eats two vectors and spits out an element of the base field, not another element of $V$, as would be the case for some sort of "product".

A very natural type of product operation that does arise on vector spaces is a Lie bracket, which generalizes things like the cross product, which really is a binary operation on your space which gives you a way to "multiply" vectors. An important feature of a Lie bracket is that it is neither commutative nor associative (but the situation isn't too bad, as it is anti-commutative and satisfies the Jacobi identity).

Answer 2

Please correct me if I'm wrong, but aren't we just talking about contraction here between two vectors? Say $$\sum_{j=1}^{n}a_{ij}y_{j}=z_{i}$$ then $$\sum_{i,j=1}^{n}a_{ij}x_i y_j=\sum_{i=1}^{n}z_i x_i$$ which is an operation between two vectors. So I guess once again it depends on the sense of the word "product" for this case as stated before, but from what's been written here it looks like this maps to $F$, not $F^n$.

Does this help?