Unit 6
Vector Spaces
Introduction
In this unit we formalize the idea of a vector space and the concept of dimension. Vector spaces and
subspaces are introduced in this unit as generalizations of the two-dimensional space
R
2
and the
three-dimensional space
R
3
. Given a vector space, sets of vectors within the space are examined to
determine if they are linearly independent and if they span the given vector space. If the set of
vectors is both linearly independent and spans the vector space it is called a
basis
. The number of
vectors in a basis is known as the
dimension
of the vector space. By the end of this unit you will
understand that the set of all ordered
m
-tuples form a vector space which is
m
dimensional and that
any real
m
-dimensional vector space can be identified with
R
m
.
A note on dimension
Most people are aware that we live in a three-dimensional world and that a sheet of paper is a two-
dimensional object. They may also know that a line is one-dimensional object and a point is a zero-
dimensional object. However, most people would be unable to give a satisfactory definition of what
the term dimension means. In a crude sense, dimension is a measure of how well an object fills
space
−
the more space an object fills, the higher is its dimension. In this unit a mathematical
treatment of the concept of dimension is undertaken and the existence of spaces having four or
more dimensions is illustrated.
Learning objectives
Upon completion of this unit you should be able to:
•
define the term “vector space”;
•
determine whether a given set is a vector space;
•
define the term “subspace”;
•
determine whether a subset of a vector space is a subspace;
•
define the term “linear combination”;
•
define the term “spanning set”;
•
determine whether a set of vectors is a spanning set of a vector space;
•
define the terms “linear dependence” and “linear independence”;
•
determine whether a set of vectors is linearly dependent or linearly independent;
•
define the term “basis”;
•
define the term “dimension of a vector space”;
•
determine the dimension of a vector space;
•
determine the coordinates of a vector relative to a given basis;
•
find a basis for a vector space;
•
define the terms “row space” and “column space” of a matrix;
•
define the term “rank of a matrix”;
•
find a basis for the row space of a matrix;
•
find a basis for the column space of a matrix;
•
determine the rank of a matrix;
•
define the terms “null space” and “nullity” for a matrix; and
•
find the null space and the nullity of a matrix.
Vector Geometry and Linear Algebra
MATH 1300
Unit 6
1

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Unit activities
1.
Although there is no assignment to be submitted in this unit, a sample set of problems with
solutions is provided so that you can undertake a self assessment on the material from this unit.