MATH 304, Sections 501 and 503
Handout 01
Did Monday September 7, 2015
Chapter 2: Vector Spaces
Section 1: Vector Spaces
In physics, one is often told a vector is an object with a magnitude and a direction. However, these
are not the defining properties of a vector. What is more important is the operations one can perform
with vectors. Specifically, one can add two vectors together and scale vectors by a number.
There are several objects in mathematics that have addition and scalar multiplication. Here we make a
formal definition of such structures which impose desired properties on addition and scalar multiplication.
Recall in this course that a field F is either Q, R, or C.
Definition 1. A vector space is a set V together with a field F and two operations +, · where for all
~v , w
~ ∈ V there is a unique element ~v + w
~ ∈ V and for each ~v ∈ V and a ∈ F there is a unique element
a · ~v ∈ V such that the following eight properties hold:
• (VS1) ~v + w
~ =w
~ + ~v for all ~v , w
~ ∈V.
• (VS2) (~v + w)
~ + ~z = ~v + (w
~ + ~z) for all ~v , w,
~ ~z ∈ V .
• (VS3) There exists a vector ~0 ∈ V such that ~v + ~0 = ~v for all ~v ∈ V .
• (VS4) For each ~v ∈ V there exists a vector −~v ∈ V such that ~v + (−~v ) = ~0.
• (VS5) a · (~v + w)
~ = (a · ~v ) + (a · w)
~ for all ~v , w
~ ∈ V and a ∈ F.
• (VS6) (a + b) · ~v = (a · ~v ) + (b · ~v ) for all ~v ∈ V and a, b ∈ F.
• (VS7) (ab) · ~v = a · (b · ~v ) for all ~v ∈ V and a, b ∈ F.
• (VS8) 1 · ~v = ~v for all ~v ∈ V .
The elements of V are called vectors, the elements of F are called scalars, + is called addition, · is called
multiplication, the vector ~0 from (VS3) is called the zero vector, and the vector −~v in (VS4) is called the
additive inverse of ~v .
Note that there are NO axioms for multiplying two vectors (i.e. you CANNOT multiply two vectors).
Subtraction will be defined using additive inverses.