CHAPTER 1: Vectors in
R
n
Section 1.1: Vectors in
R
n
For any positive integer
n
the set of all elements of the form (
x
1
,...,x
n
) where
x
i
∈
R
Definition
R
n
for 1
≤
i
≤
n
is called
n
-dimensional Euclidean space and is denoted
R
n
. The elements
of
R
n
are called
points
and are usually denoted
P
(
x
1
,...,x
n
).
2-dimensional and 3-dimensional Euclidean space was originally studied by the ancient
Greek mathematicians. In Euclid’s Elements, Euclid defines two and three dimensional
space with a set of postulates, definitions and common notions. However, in modern
mathematics, we not only want to make the concepts in Euclidean space more mathe-
matically precise, but we want to be able to easily generalize the concepts to allow us
to use the same ideas to solve problems in other areas.
In Linear Algebra, we will view
R
n
as a set of
vectors
rather than as a set of points.
In particular, we will write an element of
R
n
as a column vector and denote it with the
usual vector symbol
vectorx
(Note that some textbooks, like the one we’re using, will denote
vectors in bold face i.e.
x
). That is,
vectorx
∈
R
n
can be represented as
vectorx
=
x
1
.
.
.
x
n
,
x
i
∈
R
,
1
≤
i
≤
n.
For example, in 3-dimensional Euclidean space, we will denote the origin (0
,
0
,
0), by
the vector
vector
0 =
0
0
0
.
Imporatant Remarks:
1. Although we are going to view
R
n
as a set of vectors, it is important to understand
that we really are still referring to
n
-dimensional Euclidean space. In some cases it
will be useful to think of the vector
x
1
.
.
.
x
n
as the point (
x
1
,...,x
n
). In particular,
we will sometimes interpret a set of vectors as a set of points to get a geometric
object (such as a line or plane).
2. In Linear Algebra all vectors in
R
n
should be written as column vectors, however,
we will sometimes write these as
n
-tuples to match the notation that is commonly
used in other areas. In particular, when we are dealing with functions of vectors,
we will write
f
(
x
1
,...,x
n
) rather than
f
x
1
.
.
.
x
n
.

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2
Chapter 1
Vectors in
R
n
One advantage in viewing the elements of
R
n
as vectors instead of as points is that
we can perform operations on vectors. You likely have seen vectors in
R
2
and
R
3
in
Physics being used to represent motion or force. From these physical examples we can
observe that we add vectors by summing their components and multiply a vector by
a scalar by multiplying each entry of the vector by the scalar. We keep this definition
for vectors in
R
n
.
Let
vectorx
=
x
1
.
.
.
x
n
and
vector
y
=
y
1
.
.
.
y
n
be two vectors in
R
n
and
c
∈
R
. We define
Definition
Vector Addition
and Scalar
Multiplication
vectorx
+
vector
y
=
x
1
+
y
1
.

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