Monday February 23
−
Lecture 19 :
Coordinate of a vector with respect to a basis.
.
(Refers to section 4.4)
Expectations:
1.
Recall: When expressing a vector
v
in
V
as a linear combination of the vectors in a
basis, the coefficients are unique.
2.
Give a graphic representation of vectors for different bases in
R
2
and
R
3
.
3.
Define and find the coordinates of
v
with respect to a given basis of a vector space
V
.
19.1
Recall
−
The unique representation theorem
. If
V
is a vector space and
B
= {
v
1
,
v
2
,
....
,
v
d
}
forms a basis for
V
then for every
v
in
V
there is a
unique
set of
coefficients
α
1
, α
2,
...., α
d
in
R
such that
v
=
α
1
v
1
+
α
2
v
2
+ .
... +
α
d
v
d
.
Proof (Outline)
±
Suppose
v
=
α
1
v
1
+
α
2
v
2
+
.... +
α
d
v
d
=
β
1
v
1
+
β
2
v
2
+ .
... +
β
d
v
d
.
±
Then
0
=
(
α
1
−
β
1
)
v
1
+ (
α
2
−
β
2
)
v
2
+ .
... + (
α
d
− β
d
)
v
d
±
Since
v
1
,
v
2
,
....
,
v
d
are linearly independent then
α
i
−
β
i
= 0 for all
i.
19.2
Definition
−
If
v
is a vector in the vector space
V
which has basis
B
= {
v
1
,
v
2
,
....
,
v
d
}
(which we will call the
B
basis of
V
, so that we can distinguish it from other bases) then
the unique coefficients
α
1
, α
2,
...., α
d
used to express
v
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 Winter '08
 All
 Linear Algebra, Vectors, basis, coordinates

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