Math, asked by Rasha1344, 1 month ago

"A linear transformation T : R2 -> R2 first reflects points through the x1-axis and then reflects points through the line x2 =- x1. Show that T can also be described as a linear transformation that rotates points about the origin. What is the angle of that rotation?"

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Answered by bhunikasharma

Step-by-step explanation:

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Assume that T is linear transformation. Find the matrix of T. a) T:R2 → R2 first rotates points through −3π4 radians (clockwise) and then reflects points through the horizontal x1-axis. b) T:R2 → R2 first reflects points through the horizontal0 x1-axis and then reflects points through the line x1=x2. Show that this transformation is merely a rotation about the origin. What is the angle of the rotation? I am unsure where to start with this since i am new with this, so could anyone explain to me how to solve this? Thanks

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Answered by Afreenakbar

Answer and Explanation:

The standard basis vectors for R2 will be designated as e1 = [1, 0] and e2 = [0, 1].

The linear transformation R1 described by: can therefore be used to depict the initial reflection through the x1-axis.

R1(e1) = [-1, 0]

R1(e2) = [0, 1]

The linear transformation R2 described by: can also be used to depict the reflection through the line x2 = -x1.

R2(e1) = [0, 1]

R2(e2) = [1, 0]

We can now easily multiply the matrices that represent these linear transformations to determine the composition of these two reflections. This is,

[T] = [R2][R1]

in where [T] is the matrix corresponding to the composite linear transformation T.

When we multiply the matrices, we obtain:

[T] = [R2][R1] =[[0, 1], [1, 0]]

[[-1, 0], [0, 1]] =[[-1, 0], [0, 1]] [[0, 1], [1, 0]]

=[[-1, 0], [0, 1]] [[0, 1], [1, 0]]

=[[0, -1], [1, 0]]

This matrix [T] represents a rotation of points in R2 counterclockwise by an angle of π/2 radians (or 90 degrees) about the origin.

As a result, the linear transformation T, which first reflects points through the x1-axis and then reflects points through the line x2= -x1, can alternatively be referred to as a linear transformation that rotates points about the origin by an angle of π/2 radians (or 90 degrees).