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As far as I've seen, there are only two types of vector multiplication defined, dot product/inner product which is defined in a vector space of any dimension, even infinite, and cross product, which is only defined in three dimensions. Are there more definitions of vector products?

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Yes, one particular example is the notion of a $K$-algebra, where $K$ is a field. A $K$-algebra $A$ is a $K$-vector space $V$ along with a bilinear product $\cdot :A\times A\rightarrow A$. That is, $\cdot$ satisfies $$v\cdot (\lambda w+\mu u)=\lambda(v\cdot w) + \mu(v\cdot u)$$ $$(\lambda v+\mu w)\cdot u= \lambda (v\cdot u)+\lambda (w\cdot u)$$

  • A familiar example is $K[x]$, the $K$-vector space of all polynomials in $x$. Here, the bilinear product is polynomial multiplication.

  • Another example: given a group $G$, we can define the group algebra $KG$ as the free vector space $\mathcal{F}(G)$ with bilinear product defined by $$\left(\sum_{g\in G}{\alpha_g g}\right)\left( \sum_{h\in G}{\beta_h h}\right)=\sum_{g,h\in G}{\alpha_g\beta_h (gh)} $$ where the $gh$ is the product of $g$ and $h$ in $G$.

  • (Thanks to symplectomorphic for the suggestion) The algebra of endomorphisms $\operatorname{End}(V)$ of a $K$-vector space $V$ is the $K$-vector space of linear maps $V\rightarrow V$ with the bilinear product being given by function composition.

  • The tensor algebra $T(V)$, where the bilinear product $\otimes: T(V)\times T(V)\rightarrow T(V)$ is defined on elementary tensors by $$(v_1\otimes \cdots \otimes v_n)\otimes (w_1\otimes \cdots \otimes w_n)=v_1\otimes \cdots \otimes v_n\otimes w_1\otimes \cdots \otimes w_n$$ and extended (bi)linearly.

  • There is also the exterior algebra $\Lambda(V)$ and the symmetric algebra $\operatorname{Sym}(V)$ defined likewise to the tensor algebra.

  • More examples include $\mathbb{C}$ as a $\mathbb{R}$-vector space, the quaternions $\mathbb{H}$, the octonions $\mathbb{O}$, and the sedenions $\mathbb{S}$.

  • The cross product on $\mathbb{R}^3$ (and likewise for $\mathbb{R}^7$) also makes it into a $\mathbb{R}$-algebra.

  • The $K$-vector space of $n\times m$ matrices with entries in $K$ is a $K$-algebra with the bilinear product given by the Hadamard product (i.e. component-wise multiplication is another $K$-algebra.)

$K$-algebras show up in many fields, including Lie Algebra (concerning certain kinds of algebras, called, surprise surprise, Lie algebras), Representation Theory, and Algebraic Geometry (and linear algebra, of course), to name a few.

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  • $\begingroup$ Perhaps worth mentioning the algebra of endomorphisms on a given vector space, to show how naturally algebras arise in linear algebra. $\endgroup$ – symplectomorphic Mar 2 '15 at 1:48
  • $\begingroup$ @symplectomorphic Very true, I'll add that example in there. $\endgroup$ – Hayden Mar 2 '15 at 1:51

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