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.
Apply the transformation T (x, y) = (0.8x − 0.6y, 0.6x + 0.8y) to the scalene triangle whose vertices are (0, 0), (5, 0), and (0, 10). What kind of isometry does T seem to be? Be as specific as you can, and provide numerical evidence for your conclusion.
Suppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed by a x-shear of 1. Find the matrix A that induces T.
A=?
Consider the linear transformation T : R2[x] → R2[x] given by T(a + bx + cx2 ) = (a − b − 2c) + (b + 2c)x + (b + 2c)x2
1) Is T cyclic?
2) Is T irreducible?
3) Is T indecomposable?
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