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3.3 Coordinatization of Vectors
1
Chapter 3.
Vector Spaces
3.3
Coordinatization of Vectors
Defnition.
An
ordered basis
(
~
b
1
,
~
b
2
,...,
~
b
n
) is an “ordered set” of vec
tors which is a basis for some vector space.
Defnition 3.8.
If
B
=(
~
b
1
,
~
b
2
~
b
n
) is an ordered basis for
V
and
~v
=
r
1
~
b
1
+
r
2
~
b
2
+
···
+
r
n
~
b
n
, then the vector [
r
1
,r
2
,...,r
n
]
∈
R
n
is the
coordinate vector of
relative to
B
, denoted
B
.
Example.
Page 211 number 6.
Note.
To Fnd
B
:
(1)
write the basis vectors as column vectors to form [
~
b
1
,
~
b
2
~
b
n

],
(2)
use GaussJordan elimination to get [
I
B
]
.
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2
Defnition.
An
isomorphism
between two vector spaces
V
and
W
is a
onetoone and onto function
α
from
V
to
W
such that:
(1)
if
~v
1
,~v
2
∈
V
then
α
(
1
+
2
)=
α
(
1
)+
α
(
2
)
,
and
(2)
if
∈
V
and
r
∈
R
then
α
(
r~v
rα
(
)
.
If there is such an
α
,then
V
and
W
are
isomorphic
, denoted
V
∼
=
W
.
Note.
An isomorphism is a onetoone and onto linear transformation.
Theorem. The Fundamental Theorem o± Finite Dimensional
Vectors Spaces.
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of MichiganDearborn.
 Fall '04
 IgorDolgachev
 Vectors, Vector Space

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