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Concept# Dimension (vector space)

Summary

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.
For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite.
The dimension of the vector space V over the field F can be written as \dim_F(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written.
Examples
The vector space \R^3 has
\left{\begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} , \begin{pmatrix} 0 \ 1 \

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We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann's algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type. (C) 2011 Elsevier B.V. All rights reserved.

In previous work, we defined the category of functors F-quad, associated to F-2-vector spaces equipped with a nondegenerate quadratic form. In this paper, we define a special family of objects in the category F-quad, named the mixed functors. We give the complete decompositions of two elements of this family that give rise to two new infinite families of simple objects in the category F-quad.

2007Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH (0)(A) and HH (0)(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.