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Encyclopedia > Graded vector space

In mathematics, a graded vector space is a vector space with an extra piece of structure, known as a grading. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... The fundamental concept in linear algebra is that of a vector space or linear space. ...


Graded vector spaces

A graded vector space is a vector space V which can be written as a direct sum of the form In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

where each Vn is a vector space. For a given n the elements of Vn are then called homogeneous elements of degree n.

Graded vector spaces are common. For example the set of all polynomials in one variable form a graded vector space, where the homogeneous elements of degree n are exactly the polynomials of degree n. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

I-graded vector spaces

I-graded vector spaces generalize graded vector spaces. Let I be a set. An I-graded vector space V is a vector space that can be written as a direct sum of subspaces indexed by I: The notion of a set is one of the most important and fundamental concepts in modern mathematics. ...


A graded vector space, as defined above, is just an N-graded vector space, where N is the set of natural numbers. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

The case when I=Z2 is particularly important in physics. A Z2-graded vector space also known as a supervector space. Wikibooks Wikiversity has more about this subject: School of Physics sci. ...

If I is a semigroup, then the tensor product of two I-graded vector spaces V and W is another I-graded vector space, In mathematics, a semigroup is a set with an associative binary operation on it. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

See also

  Results from FactBites:
Super vector space - Wikipedia, the free encyclopedia (567 words)
The study of super vector spaces and their generalizations is sometimes called super linear algebra.
A homogeneous subspace of a super vector space is a linear subspace that is spanned by homogeneous elements.
Direct sums of super vector spaces are constructed as in the ungraded case with the grading given by
Outer product (409 words)
By virtue of properties (1) and (2), the vector space becomes an algebra, and by property (4) is also associative.
A vector can be seen as a "piece" of a straight line with an orientation; a bivector is a piece of a plane with an orientation.
If we took our vectors from an n-dimensional vector space, then we cannot get more than n LI vectors; thus, the outer product of more than n vectors is always 0, and the n-vector is the "highest order" k-vector that can be generated.
  More results at FactBites »



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