A First Course in Linear Algebra, 2017a

450 Vector Spaces
Definition 9.2: Axioms of Addition
Let⃗v,⃗w,⃗z be vectors in a vector space V . Then they satisfy the following axioms of addition:
• Closed under Addition
If⃗v,⃗w are in V, then⃗v + ⃗w is also in V .
• The Commutative Law of Addition
⃗v + ⃗w = ⃗w +⃗v
• The Associative Law of Addition
(⃗v +⃗w)+⃗z =⃗v +(⃗w +⃗z)
• The Existence of an Additive Identity
⃗v +⃗0 =⃗v
• The Existence of an Additive Inverse
⃗v +(−⃗v)=⃗0
Definition 9.3: Axioms of Scalar Multiplication
Let a,b ∈ R and let⃗v,⃗w,⃗z be vectors in a vector space V . Then they satisfy the following axioms of
scalar multiplication:
• Closed under Scalar Multiplication
If a is a real number, and⃗v is in V, then a⃗v is in V .
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a(⃗v + ⃗w)=a⃗v + a⃗w
(a + b)⃗v = a⃗v + b⃗v
a(b⃗v)=(ab)⃗v
1⃗v =⃗v
Consider the following example, in which we prove that R n is in fact a vector space.