If T ( x → ) = A x → is an invertible linear transformation from ℝ 2 to ℝ 2 , then the image T ( Ω ) of the unit circle Ω is an ellipse. See Exercise 2.2.54. a. Sketch this ellipse when A = [ p 0 0 q ] where p and q are positive. What is its area? b. For an arbitrary invertible transformation T ( x → ) = A x → denote the lengths of the semimajor and semi minor axes of T ( Ω ) by a and b, respectively. What is the relationship among a, b, and det ( A ) ? c. For the transformation T ( x → ) = [ 3 1 1 3 ] x → sketch this ellipse and determine its axes. Hint: Consider T [ 1 1 ] and T [ 1 − 1 ] .
If T ( x → ) = A x → is an invertible linear transformation from ℝ 2 to ℝ 2 , then the image T ( Ω ) of the unit circle Ω is an ellipse. See Exercise 2.2.54. a. Sketch this ellipse when A = [ p 0 0 q ] where p and q are positive. What is its area? b. For an arbitrary invertible transformation T ( x → ) = A x → denote the lengths of the semimajor and semi minor axes of T ( Ω ) by a and b, respectively. What is the relationship among a, b, and det ( A ) ? c. For the transformation T ( x → ) = [ 3 1 1 3 ] x → sketch this ellipse and determine its axes. Hint: Consider T [ 1 1 ] and T [ 1 − 1 ] .
Solution Summary: The author calculates the area of the given ellipse using pqpi .
If
T
(
x
→
)
=
A
x
→
is an invertible linear transformation from
ℝ
2
to
ℝ
2
, then the image
T
(
Ω
)
of the unit circle
Ω
is an ellipse. See Exercise 2.2.54. a. Sketch this ellipse when
A
=
[
p
0
0
q
]
where p and q are positive. What is its area? b. For an arbitrary invertible transformation
T
(
x
→
)
=
A
x
→
denote the lengths of the semimajor and semi minor axes of
T
(
Ω
)
by a and b, respectively. What is the relationship among a, b, and
det
(
A
)
?
c. For the transformation
T
(
x
→
)
=
[
3
1
1
3
]
x
→
sketch this ellipse and determine its axes. Hint: Consider
T
[
1
1
]
and
T
[
1
−
1
]
.
Two-dimensional figure measured in terms of radius. It is formed by a set of points that are at a constant or fixed distance from a fixed point in the center of the plane. The parts of the circle are circumference, radius, diameter, chord, tangent, secant, arc of a circle, and segment in a circle.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY