Vector Space
2.5
2001/08/09
2002/08/02 00:00:00.005 GMT-5
Doug
Daniels
rainking@alumni.rice.edu
Steven
Cox
cox@rice.edu
Doug
Daniels
rainking@alumni.rice.edu
Mark
Barrett
markb@alumni.rice.edu
Steven
Cox
cox@rice.edu
vector
space
spaces
complex
arithmetic
linear algebra
This module discusses vector spaces and their applications to complex arithmetic.
Introduction
You have long taken for granted the fact that the set of real
numbers, , is closed under addition
and multiplication, that each number has a unique additive
inverse, and that the commutative, associative, and
distributive laws were right as rain. The set,
, of complex numbers also
enjoys each of these properties, as do the sets
ℝ
n
and
ℂ
n
of columns of
n
real and complex numbers, respectively.
To be more precise, we write
x
and y
in
ℝ
n
as
x
x
1
x
2
…
x
n
y
y
1
y
2
…
y
n
and define their vector sum as the elementwise sum
x
y
x
1
y
1
x
2
y
2
⋮
x
n
y
n
and similarly, the product of a complex scalar,
z
with
x as:
z
x
z
x
1
z
x
2
⋮
z
x
n
Vector Space
These notions lead naturally to the concept of vector space. A set
V
is said to be a vector space if
-
x
y
y
x
for each
x
and
y
in
V
-
x
y
z
x
y
z
for each
x
,
y
, and
z
in
V
-
There is a unique "zero vector" such that
x
0
x
for each
x
in
V
- For each
x
in
V
there is a unique vector
x
such that
x
x
0
.
-
1
x
x
-
c
1
c
2
x
c
1
c
2
x
for each
x
in
V
and
c
1
and
c
2
in
.
-
c
x
y
c
x
c
y
for each
x
and
y
in
V
and
c
in
.
-
c
1
c
2
x
c
1
x
c
2
x
for each
x
in
V
and
c
1
and
c
2
in
.