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 *V*_{n} is a vector space. For a given *n* the elements of *V*_{n} 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. ...
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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*=**Z**_{2} is particularly important in physics. A **Z**_{2}-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 |