Linear Algebra Basics
A vector space (or, linear space) is an algebraic structure that often provides a home for solutions of mathematical models.
Linear algebra
is a study of vector spaces and a class of special functions (called “linear transformations”) between them.
The terms
vector space
and
linear space
are interchangeable. We’ll use them both.
1.
Linear Spaces
: The elements of a linear space are called vectors.
We can scale and add vectors.
By scaling and
adding vectors we can build new vectors. This simple construction paradigm is remarkably useful.
Definition
: A
linear space
(or
vector space
) is a set
L
together with field of
scalars
F
and two functions (operations);
addition + :
L
×
L
→
L
and (scalar) multiplication
·
:
F
×
L
→
L
satisfying conditions (a)(h) below.
For us, the scalar field
F
will almost always be the real numbers
R
(rarely, for us, it may be the complex numbers
C
).
Also, for brevity (clarity?), if
α
∈
F
and
x
∈
L
and we want to scale
x
using
α
, then we’ll write
αx
rather than
α
·
x
.
(a)
x
+
y
=
y
+
x
for
x
,
y
in
L
(b)
x
+ (
y
+
z
) = (
x
+
y
) +
z
for
x, y, z
∈
L
(c) there is an element
O
∈
L
such that for each
x
∈
L
,
x
+
O
=
x
(d) for each
x
∈
L
there corresponds
y
∈
L
such that
x
+
y
=
O
(e) for each
α, β
∈
F
and
x
∈
L
,
α
(
βx
) = (
αβ
)
x
(f) for each
α, β
∈
F
and
x
∈
L
, (
α
+
β
)
x
=
αx
+
βx
(g) for each
α
∈
F
and
x, y
∈
L
,
α
(
x
+
y
) =
αx
+
αy
, and
(h) 1
x
=
x
for each
x
∈
L
.
Remarks
:
•
If
x
+
y
=
x
+
z
then
y
=
z
. This
cancellation
law holds because
y
=
O
+
y
= (

x
+
x
) +
y
=

x
+ (
x
+
y
) =

x
+ (
x
+
z
) = (

x
+
x
) +
z
=
O
+
z
=
z
.
•
0
x
+
x
= (0 + 1)
x
=
x
so 0
x
=
O
.
•
The zero element
O
of (b) and
y
, the additive inverse of
x
, in (c) are unique; we usually write

x
for the additive
inverse of
x
. Since

1
x
+
x
=

1
x
+ 1
x
= (

1 + 1)
x
= 0
x
=
O
then

x
=

1
x
.
•
For us, the scalar field
F
will always be either the real numbers
R
or complex numbers
C
. There are other fields
however, for example the rational numbers
Q
. There are finite fields, e.g.,
Z
p
(the integers mod
p
) where
p
is a
prime number. Vector spaces over finite fields find applications in, for example, crytography. We’ll not consider
them in this course.
Examples of linear spaces:
(a)
R
n
: A vector
x
∈
R
n
is an
n
tuple of real numbers,
x
= (
x
1
, x
2
, . . . , x
n
). The set of real numbers
R
is the scalar
field.
Vector additon and scalar multiplication are defined componentwise: if
x
and
y
are vectors and
α
∈
R
then
x
+
y
= (
x
1
, x
2
, . . . , x
n
) + (
y
1
, y
2
, . . . , y
n
) := (
x
1
+
y
1
, x
2
+
y
2
, . . . , x
n
+
y
n
) and
αx
=
α
(
x
1
, x
2
, . . . , x
n
) :=
(
αx
1
, αx
2
, . . . , αx
n
).
(b)
C
n
:
A vector
z
∈
C
n
is an
n
tuple of complex numbers,
z
= (
z
1
, z
2
, . . . , z
n
).
The set of complex numbers
C
is the scalar field.
Vector addition and scalar multiplication are defined componentwise:
if
w
and
z
are
vectors and
α
∈
C
then
w
+
z
= (
w
1
, w
2
, . . . , w
n
) + (
z
1
, z
2
, . . . , z
n
) := (
w
1
+
z
1
, w
2
+
z
2
, . . . , w
n
+
z
n
) and
αz
=
α
(
z
1
, z
2
, . . . , z
n
) := (
αz
1
, αz
2
, . . . , αz
n
).
(c)
R
n
p
: If
p
∈
R
n
then define
R
n
p
:=
{
(
p, x
)

x
∈
R
n
}
.
R
n
p
(vectors based at
p
) becomes a real vector space if addition
and scalar multiplication are defined by (
p, x
) + (
p, y
) = (
p, x
+
y
) and
α
(
p, x
) = (
p, αx
) for each scalar
α
∈
R
and
pair of vectors
x, y
∈
R
n
.
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(d)
M
m
×
n
: A vector is a rectangular
m
×
n
array of real numbers.
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 Spring '09
 Linear Algebra, Vectors, Vector Space, vk, Mm×n

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