Linear Algebra

Section I. Definition of Vector Space 83Example 1.15 has brought us full circle since it is one of our motivatingexamples. Now, with some feel for the kinds of structures that satisfy thedefinition of a vector space, we can reflect on that definition. For example, whyspecify in the definition the condition that 1 · ⃗v = ⃗v but not a condition that0 · ⃗v = ⃗0?One answer is that this is just a definition — it gives the rules of the gamefrom here on, and if you don’t like it, move on to something else.Another answer is perhaps more satisfying. People in this area have workedhard to develop the right balance of power and generality. This definition isshaped so that it contains the conditions needed to prove all of the interestingand important properties of spaces of linear combinations. As we proceed, weshall derive all of the properties natural to collections of linear combinationsfrom the conditions given in the definition.The next result is an example. We do not need to include these propertiesin the definition of vector space because they follow from the properties alreadylisted there.1.16 Lemma In any vector space V, for any ⃗v ∈ V and r ∈ R, we have (1) 0·⃗v = ⃗0,and (2) (−1 · ⃗v) + ⃗v = ⃗0, and (3) r · ⃗0 = ⃗0.Proof For (1), note that ⃗v = (1 + 0) · ⃗v = ⃗v + (0 · ⃗v). Add to both sides theadditive inverse of ⃗v, the vector ⃗w such that ⃗w + ⃗v = ⃗0.⃗w + ⃗v = ⃗w + ⃗v + 0 · ⃗v⃗0 = ⃗0 + 0 · ⃗v⃗0 = 0 · ⃗vItem (2) is easy: (−1 ·⃗v) +⃗v = (−1 + 1) ·⃗v = 0 ·⃗v = ⃗0 shows that we can write‘−⃗v ’ for the additive inverse of ⃗v without worrying about possible confusionwith (−1) · ⃗v.For (3) r · ⃗0 = r · (0 · ⃗0) = (r · 0) · ⃗0 = ⃗0 will do.QEDWe finish with a recap. Our study in Chapter One of Gaussian reductionled us to consider collections of linear combinations. So in this chapter we havedefined a vector space to be a structure in which we can form such combinations,expressions of the form c 1 · ⃗v 1 + · · · + c n · ⃗v n (subject to simple conditions onthe addition and scalar multiplication operations). In a phrase: vector spacesare the right context in which to study linearity.Finally, a comment. From the fact that it forms a whole chapter, andespecially because that chapter is the first one, a reader could suppose thatour purpose is the study of linear systems. The truth is, we will not so muchuse vector spaces in the study of linear systems as we will instead have linearsystems start us on the study of vector spaces. The wide variety of examplesfrom this subsection shows that the study of vector spaces is interesting andimportant in its own right, aside from how it helps us understand linear systems.