Notes on Linear Algebra
January 18, 2010
These notes are intended to give a brief overview of some key terms and
ideas from linear algebra, without all the technical details. For full details
see any linear algebra textbook.
1
Vector spaces, inner products
A vector space
V
is a set of objects that can be added to one another or
multiplied by any scalar belonging a given number field
F
, to yield another
object in
V
. Usually
F
=
R
or
F
=
C
(the complex numbers), although in
principle
F
could be another field, such as the rational numbers. For present
purposes, let’s take the scalar field
F
to be the real numbers
R
.
Here are
some examples of vector spaces.
1.
V
=
R
3
. This serves as a model for ordinary space. We denote elements
x, y
∈
R
3
by a column vectors
x
=
x
1
x
2
x
3
,
y
=
y
1
y
2
y
3
,
x
+
y
=
x
1
+
y
1
x
2
+
y
2
x
3
+
y
3
.
Given any scalar
α
∈
R
, the product
αx
∈
R
3
is a vector.
2. A more general family of examples is
V
=
R
n
, where
n
≥
1 is any fixed
natural number. If
n
= 1 we have the real numbers themselves as an
example of a vector space, and if
n
= 3 we have the previous example.
1
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For general
n
, we still represent elements
x, y
∈
R
n
as column vectors
x
=
x
1
x
2
.
.
.
x
n
,
y
=
y
1
y
2
.
.
.
y
n
,
x
+
y
=
x
1
+
y
1
x
2
+
y
2
.
.
.
x
n
+
y
n
.
Scalar multiplication works as in the previous example: for any
α
∈
R
and
x
∈
R
n
,
αx
=
αx
1
αx
2
.
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 Winter '10
 PeterGibson
 Differential Equations, Linear Algebra, Algebra, Equations, Vector Space, αj, αx

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