This chapter lays the ground work for one of the main features of linear algebra: vector spaces. It defines the motivation for them, and then gives formal definitions. Over the course of the chapter, several important theorems are discussed.
The motivation of vector spaces comes from a natural extension of the fields we care about: R and C. When we think about addition in R, it’s an operation that combines two elements of R in a commutative, associative way, and that has an additive identity.
We try to extend that to lists of elements from R, and other entities that turn out to behave very similarly.
Important Definitions A vector space is a set V with an addition and scalar multiplication with the following properties: commutativity ${latex.inlineu + v = v + u\quad \forall u, v \in V}
associativity \({latex.inline[(u + v) + w = u + (v + w)]((u + v) + w = u + (v + w))} and \){latex.inline(ab)v = a(bv)} where u,v,w come from V and a, b come from F.
additive identity ${latex.inline\exists 0 \in V\ :\ v + 0 = v\quad \forall v \in V}
additive inverse ${latex.inline\forall v \in V\ \exists w \in V\ : v + w = 0}
multiplicative identity ${latex.inline1v = v\quad \forall v \in V}
distributive property follows from above, works as expected
Relevant Theorems * 1753094499 - Axler 1.26 Unique additive identity * 1753094531 - Axler 1.27 Unique additive inverse * 1753094549 - Axler 1.30 0 times vector is vector 0 * 1753094549 - Axler 1.31 Any scalar times 0 vector is 0 vector * 1753094630 - Axler 1.32 -1 * v = -v