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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?


$F$ 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!

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    @JonasMeyer: Thanks! I was thinking maybe they call it tensor product or something defined in another way? Do they?2012-01-16

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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).

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    And I was maybe being overly pedantic... ;) I think it would be productive at this point to pin down what you mean by "product", as I'm still not entirely clear on this. There is a sense in which the answer to your question is "trivially yes", which the following: if by a "product" you mean a map $V \times V \rightarrow W$ (or some generalization) that gives you a way of combining two elements of $V$ and getting some element of $W$ (all vector spaces), then the answer is "trivially yes", because we only ever care about (multi)-linear maps between vector spaces.2012-01-17
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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?

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    Sure, I've always thought o$f$ it as essentially the "dot product", but it is the multiplication between the same components of n-tuples, summed. Generally I see it in physics with the "summation convention" between vectors and co-vectors. Is this clear?2012-01-17