Multilinear formIn abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map that is separately -linear in each of its arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces. A multilinear -form on over is called a (covariant) -tensor, and the vector space of such forms is usually denoted or .
TransposeIn linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT.
Bilinear formIn mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on is an example of a bilinear form.
Multilinear mapIn linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function where and are vector spaces (or modules over a commutative ring), with the following property: for each , if all of the variables but are held constant, then is a linear function of . A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of k variables is called a k-linear map.
Vector spaceIn mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.
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