ABSTRACT
A vector space V is just a module over a field F. Since the study of vector spaces far predated that of modules, vector space theory has its own standard terminology. We’ll use this established terminology, so here’s a glossary. A vector is an element of a vector space. We use boldface type to distinguish vectors. A subspace is just an F-submodule. We retain the notation W ≤ V to mean that W is a subspace of V. The subspace spanned by a set of vectors {v a : α ∈ A} is the F-submodule generated by this collection of vectors. We use < v a : α ∈ A > to denote this subspace. Recall that < v a : α ∈ A > is the set of all finite sums {∑ α∈A 0 fα v α: fα ∈ F, A 0 is a finite subset of A}. An expression of the form ∑ α∈A 0 fα v α with A 0 a finite subset of A is called a linear combination of the vectors {vα : α ∈ A}. We say the set of vectors {vα : α ∈ A} ⊂ V is a spanning set for V if < v a : α ∈ A >= V. If V, W are vector spaces over F, a linear transformation from V to W is an F-module homomorphism from V to W. The words isomorphism, monomorphism and epimorphism are imported from module theory to vector space theory without change, as are the definitions of kernel and image. We also retain the notation FV to indicate that V is a vector space (module) over the field F. If W ≤ V the factor space is just the factor F-module V/W. Note that a vector space FV is torsion-free as an F-module. To see this, let v ∈ V and 0 ≠ f ∈ F with f v = 0. Then v = f −1 f v = f −10 = 0. Thus, the annihilator of every 0 ≠ v ∈ V is zero; equivalently there are no nonzero torsion elements of V.
