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Let q> 1. A circle in R is defined to be the set of all points in some two-dimensional affine subspace of Rª that are a fixed positive distance from some fixed point in that affine subspace. The fixed positive distance is the radius of the circle and the fixed point is its center. Let Y, Z₁, and Z2 be members of Rª and suppose that Z₁ and Z₂ have the same norm and are perpendicular to each other. Let F(t) = Y + Z₁ cost + Z₂ sint, 0≤t<2π. Prove that the image of F is a circle by first showing that every member of the image of F is on some circle and then showing that every member of that circle belongs to the image of F. What are the center and radius of the circle?

Let q> 1. A circle in R is defined to be the set of all points in some two-dimensional affine subspace of Rª that are a fixed positive distance from some fixed point in that affine subspace. The fixed positive distance is the radius of the circle and the fixed point is its center. Let Y, Z₁, and Z2 be members of Rª and suppose that Z₁ and Z₂ have the same norm and are perpendicular to each other. Let F(t) = Y + Z₁ cost + Z₂ sint, 0≤t<2π. Prove that the image of F is a circle by first showing that every member of the image of F is on some circle and then showing that every member of that circle belongs to the image of F. What are the center and radius of the circle?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Let q> 1. A circle in R9 is defined to be the set of all points in some two-dimensional affine subspace of R9 that are a fixed positive distance from some fixed point in that affine subspace. The fixed positive distance is the radius of the circle and the fixed point is its center. Let Y, Z₁, and Z2 be members of R9 and suppose that Z₁ and Z₂ have the same norm and are perpendicular to each other. Let F(t) = Y + Z₁ cost + Z₂ sint, 0≤ t < 2π. Prove that the image of F is a circle by first showing that every member of the image of F is on some circle and then showing that every member of that circle belongs to the image of F. What are the center and radius of the circle?
Transcribed Image Text:Let q> 1. A circle in R9 is defined to be the set of all points in some two-dimensional affine subspace of R9 that are a fixed positive distance from some fixed point in that affine subspace. The fixed positive distance is the radius of the circle and the fixed point is its center. Let Y, Z₁, and Z2 be members of R9 and suppose that Z₁ and Z₂ have the same norm and are perpendicular to each other. Let F(t) = Y + Z₁ cost + Z₂ sint, 0≤ t < 2π. Prove that the image of F is a circle by first showing that every member of the image of F is on some circle and then showing that every member of that circle belongs to the image of F. What are the center and radius of the circle?
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