Linear transformation rotation

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Homework Help Overview

The discussion revolves around a linear transformation in R2 involving reflections and rotations. The original poster is trying to understand the outcome of applying a clockwise rotation of -3pi/4 radians followed by a reflection through the x1-axis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions the resulting coordinates after the transformation, specifically why they differ from their expectations based on the angle of rotation. Other participants discuss the implications of reflection and clarify the relationship between the coordinates and the unit circle.

Discussion Status

Participants are actively engaging in clarifying the transformations involved. There is a focus on understanding the effects of both the rotation and the subsequent reflection, with some guidance provided regarding the mathematical relationships between the coordinates.

Contextual Notes

There is a hint provided regarding the transformation of the standard basis vector e1, which may influence the understanding of the problem. The discussion also touches on potential confusion regarding the representation of coordinates in relation to the unit circle.

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Homework Statement


T: R2-->R2 first reflects points through -3pi/4 radian (clockwise) and then reflects points through the horizontal x1-axis. [Hint T(e1)= (-1/sqrt2, 1/sqrt2)



The Attempt at a Solution


I just don't understand why the points would be (-1/sqrt2, 1/sqrt2). If it's -3pi/4, why wouldn't it be (-(sqrt2)/2, -(sqrt2)/2)?
 
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Did you miss the part about the reflection or are you rotating the wrong way? Rotating (1, 0) -3pi/4 radians (clockwise as it says) gives you (-1/sqrt(2), -1/sqrt(2)) but then the reflection through the horizontal axis changes that to (-1/sqrt(2),+1/sqrt(2)).
 
Doesn't -3pi/4 radians correlate to (-(sqrt2)/2, -(sqrt2)/2) though (on a unit circle)? Where is the sqrt2 on the bottom coming from?
 
Are you aware that 2= (sqrt(2))*(sqrt(2))? 1/sqrt(2) is exactly the same as sqrt(2)/2.
 

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