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Any bilinear map is a multilinear map. For example, any inner product on a -vector space is a multilinear map, as is the cross product of vectors in .; The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
The determinant, permanent and other immanants of a matrix are homogeneous multilinear polynomials in the elements of the matrix (and also multilinear forms in the rows or columns). The multilinear polynomials in n {\displaystyle n} variables form a 2 n {\displaystyle 2^{n}} -dimensional vector space , which is also the basis used in the ...
In 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. [1] More generally, one can define multilinear forms on a module over a commutative ring.
Only their multilinearity with respect to rows and columns, and their alternating property (vanishing in the presence of equal rows or columns) are used; in particular the multiplicative property of determinants for square matrices is not used, but is rather established (the case n = m). The proof is valid for arbitrary commutative coefficient ...
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal.
The progressive development of this and other schools during Muir's tenure as Superintendent-General of Education was strongly influenced by Muir's belief in the importance of subjects requiring manual dexterity (woodwork and needlework) and the teaching of mathematics, science and nature study.