In the litterature we see the terminology "multilinear form" or "$n$-form". I'm used to refer the word "form" to mean a homogeneous polynomial. but here we define it as a map $f:V^n\to F$, ($V$ is an $F$-vector space), such that $f$ is linear on each of its components. I'm confused by terminology, for example is there some connection between a bilinear form and a quadratic form?
terminology of multilinear form
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linear-algebra
terminology
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0[Wikipedia discusses the relation of bilinear and quadratic forms](http://en.wikipedia.org/wiki/Quadratic_form#Definitions) – 2012-06-25
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0If you write $n-$form with the hyphen INSIDE the $\TeX$ code, then it looks like a minus sign rather than a hyphen. I changed it to $n$-form. Similarly $F-$vector and $F$-vector. – 2012-06-25
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1The dictionary between symmetric bilinear forms and quadratic forms (outside characteristic 2) that Qiaochu mentions in his answer can be extended to symmetric multilinear forms and homogeneous polynomials using a process called polarization. See http://en.wikipedia.org/wiki/Polarization_of_an_algebraic_form. – 2012-06-26