These things are equivalent to the homomorphism of a module defined over a field, rather than a ring. Because these functions preserve the origin, they are subset of affine functions where the translation is zero. The space of all of these things whose codomain is a scalar field defines the dual space of their domain. These things are (*) invertible, being both surjective 
These things are equivalent to the homomorphism of a module defined over a field, rather than a ring. Because these functions preserve the origin, they are subset of affine functions where the translation is zero. The space of all of these things whose codomain is a scalar field defines the dual space of their domain. These things are (*) invertible, being both surjective and injective, if their domain and codomain have the same dimension. The dimension of an input vector space is equal to the sums of the dimensions of the kernel and range of these things. For 10 points, name these additive and homogeneous functions that are defined between two vector spaces.
These things are equivalent to the homomorphism of a module defined over a field, rather than a ring.
linear mapping [or: linear transformation; prompt on: "linear operator", "linear function", "linear functional", "affine transformation", "module homomorphism"]