Super Vector Space - Linear Transformations

Linear Transformations

A homomorphism from one super vector space to another is a grade-preserving linear transformation. A linear transformation f : V → W between super vector spaces is grade preserving if

for i = 0 and 1. That is, it maps the even elements of V to even elements of W and odd elements of V to odd elements of W. An isomorphism of super vector spaces is a bijective homomorphism.

Every linear transformation 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 f : V → W such that

for i = 0 and 1. Declaring the grade-preserving transformations to be even and the grade-reversing ones to be odd gives the space of all linear transformations from V to W the structure of a super vector space.

Note that a grade-reversing transformation from V to W can be regarded as a homomorphism from V to the parity reversed space ΠW.