gives the space of all linear transformations from to , denoted and called internal , the structure of a super vector space. In particular, A grade-reversing transformation from to can be regarded as a homomorphism from to the parity reversed space , so that Operations on super vector spaces The usual algebraic constructions for ordinary vector spaces have their counterpart in the super vector space setting. Dual space The dual space of a super vector space can be regarded as a super vector space by taking the even functionals to be those that vanish on and the odd functionals to be those that vanish on . Equivalently, one can define to be the space of linear maps from to (the base field thought of as a purely even super vector space) with the gradation given in the previous section. Direct sum Direct sums of super vector spaces are constructed as in the ungraded case with the grading given by Tensor product One can also construct tensor products of super vector spaces. Here the additive structure of comes into play. The underlying space is as in the ungraded case with the grading given by where the indices are in . Specifically, one has Supermodules Just as one may generalize vector spaces over a field to modules over a commutative ring, one may generalize super vector spaces over a field to supermodules over a supercommutative algebra (or ring). A common construction when working with super vector spaces is to enlarge the field of scalars to a supercommutative Grassmann algebra. Given a field let denote the Grassmann algebra generated by anticommuting odd elements . Any super vector space over can be embedded in a module over by considering the (graded) tensor product The category of super vector spaces The category of super vector spaces, denoted by , is the category whose objects are super vector spaces (over a fixed field ) and whose morphisms are even linear transformations (i.e. the grade preserving ones). The categorical approach to super linear algebra is to first formulate definitions and theorems regarding ordinary (ungraded) algebraic objects in the language of category theory and then transfer these directly to the category of super vector spaces. This leads to a treatment of "superobjects" such as superalgebras, Lie superalgebras, supergroups, etc. that is completely analogous to their ungraded counterparts. The category is a monoidal category with the super tensor product