In mathematics, a super vector space is a -graded vector space, that is, a vector space over a field with a given decomposition of subspaces of grade and grade . The study of super vector spaces and their generalizations is sometimes called super linear algebra. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry. Definitions A super vector space is a -graded vector space with decomposition Vectors that are elements of either or are said to be homogeneous. The parity of a nonzero homogeneous element, denoted by , is or according to whether it is in or , Vectors of parity are called even and those of parity are called odd. In theoretical physics, the even elements are sometimes called Bose elements or bosonic, and the odd elements Fermi elements or fermionic. Definitions for super vector spaces are often given only in terms of homogeneous elements and then extended to nonhomogeneous elements by linearity. If is finite-dimensional and the dimensions of and are and respectively, then is said to have dimension . The standard super coordinate space, denoted , is the ordinary coordinate space where the even subspace is spanned by the first coordinate basis vectors and the odd space is spanned by the last . A homogeneous subspace of a super vector space is a linear subspace that is spanned by homogeneous elements. Homogeneous subspaces are super vector spaces in their own right (with the obvious grading). For any super vector space , one can define the parity reversed space to be the super vector space with the even and odd subspaces interchanged. That is, Linear transformations A homomorphism, a morphism in the category of super vector spaces, from one super vector space to another is a grade-preserving linear transformation. A linear transformation between super vector spaces is grade preserving if That is, it maps the even elements of to even elements of and odd elements of to odd elements of . An isomorphism of super vector spaces is a bijective homomorphism. The set of all homomorphisms is denoted . Every linear transformation, not necessarily grade-preserving, from one super vector space to another can be written uniquely as the sum of a grade-preserving transformation and a grade-reversing one—that is, a transformation such that Declaring the grade-preserving transformations to be even and the grade-reversing ones to be odd