This chapter introduces linear maps which is what makes linear algebra fun. In fact, at the beginning of this book, linear algebra is the study of linear maps over finite dimensional vector spaces. Chapter 2 dove into vector spaces, so all that remains is gaining a solid understanding of linear maps.
Important Definitions A linear map from V to W is a function T: V -> W with the following properties:
additivity ${latex.inlineT(u + v) = Tu + Tv\quad \forall u, v \in V}
homogeneity ${latex.inlineT(\lambda v) = \lambda (Tv)\quad \forall \lambda \in F\ and\ all\ v \in V}
As notation, we call the set of linear maps from V to W L(V, W).
Addition and product are defined by: \({latex.inline[(S + T)(v) = Sv + Tv]((S + T)(v) = Sv + Tv)} and \){latex.inline(\lambda T)(v) = \lambda(Tv)}
The product of linear maps is defined as function composition.
Relevant Theorems * 1753750954 - Axler 3.4 Linear map lemma * 1753750972 - Axler 3.6 L(V, W) is a vector space * 1753751076 - Axler 3.8 Algebraic properties of products of linear spaces * 1753751116 - Axler 3.10 Linear maps take 0 to 0