MATH1850U/2050U:
Chapter 4 cont.
..
1
GENERAL VECTOR SPACES cont.
..
Coordinates and Basis (4.4; pg. 200)
Note regarding nonrectangular coordinate systems:
A coordinate system establishes
a onetoone correspondence with all points in the plane, and we use an ordered pair to do
this.
The numbers in the ordered pair usually correspond to units along perpendicular
axis, but this is not necessary.
Any two nonparallel lines can be used to define a
coordinate system in the plane.
Note:
We can specify a coordinate system in terms of basis vectors.
Note:
For a coordinate system, we need:
Unique representation of everything in the vector space
Ability to “label” everything in the vector space
Note:
A basis can be used to define a coordinate system in a vector space.
Example:
n
e
e
e
,
,
,
2
1
are a basis for
R
n
.
Definition:
If
V
is any vector space and
n
S
v
v
v
,
,
,
2
1
is a set of vectors in
V
, then
S
is called a
basis
for
V
if the following two conditions hold:
a)
S
is linearly independent
b)
S
spans
V
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Chapter 4 cont.
..
2
Theorem (Uniqueness of Basis Representation):
If
n
S
v
v
v
,
,
,
2
1
is a basis for a
vector space
V
, then every vector
v
in
V
can be expressed in the form
n
n
c
c
c
v
v
v
v
2
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 Spring '11
 PaulaTu
 Linear Algebra, Vector Space, basis

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