Change Coordinate
C
h
a
n
g
e
C
o
o
r
d
i
n
a
t
e
Coordinate System
Other System → Cartesian
Other System → Cartesian
(matrix-vector product)
Cartesian → Other System
find [
v
]
B
B is invertible (?)
independent
•
Let
B
=
{b
1
,
b
2
,
,
b
n
}
(Standard vector)
=
c
1
b
1
+
c
2
b
2
+
+
c
n
b
n
Equation of ellipse
3
2
?
3
2
Rotate 45◦
Equation of ellipse
3
2
Use another coordinate system
What is the equation of the ellipse
in the new coordinate system?
Equation of ellipse
E
q
u
a
t
i
o
n
o
f
h
y
p
e
r
b
o
l
a
30
◦
Equation of hyperbola
30
◦
Equation of hyperbola
This content covers various topics in mathematics such as coordinate systems, vector transformations, equations of ellipses, and hyperbolas. It delves into the concepts of basis vectors, matrix-vector products, and different coordinate systems like Cartesian and other systems. Explore the methods to represent vectors and equations in these systems, including discussions on hyperbolic equations and geometric transformations.
Uploaded on Feb 16, 2025 | 0 Views
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Presentation Transcript
Change Coordinate Change Coordinate
Coordinate System Let vector set B= ?1,?2, ,?? be a basis for a subspace Rn B is a coordinate system For any v in Rn, there are unique scalars ?1,?2, ,??such that ? = ?1?1+ ?2?2+ + ???? B-coordinate vector of v: ?B= ( B ?)
Other System Cartesian 3 6 1 1 1 1 1 2 2 B= ?B= , , 1 1 2 7 1 1 1 1 1 2 2 = 7 5 ? = 3 + 6 2 1 1 3 6 7 8 9 1 2 3 4 5 6 C= ?C= , , 2 1 2 3 13 20 27 7 8 9 4 5 6 = ? = 3 + 6 2
Other System Cartesian Let vector set B= ?1,?2, ,?? be a basis for a subspace Rn Matrix B = ?1 ?2 ?? ?1 ?2 ?? Given ?B, how to find v? ?B= ? = ?1?1+ ?2?2+ + ???? = ? ?B (matrix-vector product)
Cartesian Other System 1 1 1 1 1 2 2 1 B= find [v]B , , 1 1 ? = 4 4 ?1 ?2 ?3 [v]B= 1 1 1 1 1 2 2 independent B is invertible (?) ? = 1 1 6 4 3 ?B= ? 1? = ? ?B= ?
Cartesian Other System Let B={b1 , b2 , , bn} ?B= ? 1? ? ?B ?1 ?2 ?? ? = ? ?B = = c1b1 + c2b2 + + cnbn Let B= ?1,?2, ,?? be a basis of Rn. ?? B=??? (Standard vector)
Equation of ellipse Rotate 45 2 2 3 3 2 2 x +y = 1 ? 2 2 3 2
Equation of ellipse Use another coordinate system ?2 ?1 2 2 2 2 B={ , } 2 2 2 2 2 3 ?1 ?2 What is the equation of the ellipse in the new coordinate system? 2 2 ( ) 2 ( ) 2 x y + = 1 3 2
Equation of ellipse 2 2 2 2 B={ , } 2 2 2 2 2 2 ( ) 2 ( ) 2 x y + = 1 3 2
Equation of hyperbola ? ? ? ? 30 ? ? 3?2+ 2?? + 3?2= 12 ?????????
Equation of hyperbola ? = ?1 ?2 1 ? 3 ? 2 3 2 2 1 2 ?1= ?2= ? 30 ?2 ?1 ? ? ?B=? ? = ? ?B ? = ? ? 3 1 ? ?= ? ? 2 1 2 2 3 2 3?2+ 2?? + 3?2= 12
Equation of hyperbola ? = ?1 ?2 3?2+ 2?? + 3?2= 12 1 3 2 3 2 2 1 2 ?1= ?2= 2? 1 3 2? ? = ? =1 3 ? ? 2? + 2? ?B=? ? = ? ?B ? = ? ? ? = 3 3 1 ? ?= ? ? 2 1 2 2 3 2
