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MATHEMATICS 54
Professor Constantin Teleman
9/29/09
Lecture 10
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C O P Y
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ANNOUNCEMENTS
*Midterm will not cover section 2.5 and 4.7.
*Homework for section 4.7 due on Monday.
*Eric Wayman’s 2pm section will go to an
overflow room. Watch webpage or ask in section
tomorrow.
*There is a sample exam on the course webpage.
Today we will talk about coordinate changes from
Chapter 4 around thirty minutes and then we will do
a review.
LECTURE
Coordinate Changes:
V is a finitedimensional vector space.
Assume we have chosen a finite basis
r
b
1
...
r
b
n
1
n=dim.
Recall: this defines a linear isomorphism of V with
R
n
. Aug
r
v
∈
V
has a unique expression
x
1
b
1
+
...
x
n
b
n
,
x
i
∈
R
Identification with
R
n
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
←
+
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
→
n
n
n
n
x
x
b
x
b
x
x
x
v
.
.
...
.
.
1
1
1
1
r
r
r
Given another basis D=
r
d
1
...
r
d
n
Get new coordinates on V, different identification
with
R
n
, want: a formula relating coordinates in B
with coordinates in D.
2
1
2
2
1
1
2
1
2
1
1
1
2
1
2
1
2
1
2
2
1
1
2
2
1
1
2
2
3
1
2
2
3
1
2
2
3
3
2
2
1
3
2
2
1
x
x
y
x
x
y
x
x
y
y
D
x
D
y
x
y
D
y
y
y
y
x
x
b
x
b
x
d
y
d
y
−
=
+
−
=
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
=
=
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
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This note was uploaded on 12/01/2009 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Chorin
 Math

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