Page 103 - A Course in Linear Algebra with Applications
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Chapter Four
INTRODUCTION TO VECTOR SPACES
The aim of this chapter is to introduce the reader to the
notion of an abstract vector space. Roughly speaking, a vec-
tor space is a set of objects called vectors which it is possible
to add and multiply by scalars, subject to reasonable rules.
Vector spaces occur in numerous branches of mathematics, as
well as in many applications; they are therefore of great im-
portance and utility. Rather than immediately confront the
reader with an abstract definition, we prefer first to discuss
some vector spaces which are familiar objects. Then we pro-
ceed to extract the common features of these examples, and
use them to frame the definition of a general vector space.
4.1 Examples of Vector Spaces
The first example of a vector space has a geometrical
background.
Euclidean space
Choose and fix a positive integer n, and define
to be the set of all n-column vectors
X l
f \
x = X J
\X J
n
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