Matrix Transformations from R^n to R^n

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Discussion Overview

The discussion revolves around matrix transformations, specifically focusing on rotation matrices in \(\mathbb{R}^2\). Participants explore the effects of multiplying vectors by rotation matrices and their transposes, as well as the derivation of these matrices.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about the effect of multiplying a vector \(X\) by the transpose of a rotation matrix \(A\), specifically asking for an explanation of the reasoning behind it.
  • There is a request to write out the matrix \(A(\theta)\) that represents rotation by an angle \(\theta\), with a focus on understanding its transpose.
  • One participant provides a proposed form for the rotation matrix \(A\), suggesting it as \(A = [\cos^2(\theta) - \sin^2(\theta), -2\sin(\theta)\cos(\theta); 2\sin(\theta)\cos(\theta), \cos^2(\theta) - \sin^2(\theta)]\), but this is challenged by another participant.
  • A later reply emphasizes the need to derive the rotation matrix based on its effect on the standard basis vectors, specifically noting how the vector \((1,0)\) transforms under rotation.
  • Participants discuss the mapping of the standard basis vector \((0,1)\) and reference angle-sum identities to derive the correct transformation.

Areas of Agreement / Disagreement

There is disagreement regarding the correct form of the rotation matrix \(A(\theta)\). Some participants challenge the proposed matrix and seek clarification on its derivation, indicating that the discussion remains unresolved.

Contextual Notes

Participants express uncertainty about the correct formulation of the rotation matrix and its transpose, highlighting the need for clarity on the transformations of basis vectors and the application of angle-sum identities.

Swati
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1. If multiplication by A rotates a vector X in the xy-plane through an angle (theta). what is the effect of multiplying x by A^T ? Explain Reason.
 
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Swati said:
1. If multiplication by A rotates a vector X in the xy-plane through an angle (theta). what is the effect of multiplying x by A^T ? Explain Reason.

Write out the matrix \(A(\theta)\) that rotates vectors by an angle \( \theta\). Now take its transpose, what do you notice?

CB
 
Sorry, I'm not getting it. Can you explain in brief.
 
Swati said:
Sorry, I'm not getting it. Can you explain in brief.

What is the matrix \(A(\theta)\) (the rotation matrix that rotates vectors in \(\mathbb{R}^2\) by \(\theta\) ) written out in full?

CB
 
CaptainBlack said:
What is the matrix \(A(\theta)\) (the rotation matrix that rotates vectors in \(\mathbb{R}^2\) by \(\theta\) ) written out in full?

CB

A=[cos^2(theta)-sin^2(theta), -2sin(theta)cos(theta) ;
2sin(theta)cos(theta),
cos^2(theta)-sin^2(theta)]

(A is 2*2 matrix.)
 
err...no, it's not.

suppose we rotate (counter-clockwise) through an angle of θ.

to get the matrix for such a rotation, we need to know its effect on a basis for the plane.

there's no compelling reason not to use the standard basis {(1,0),(0,1)}, so we will.

it should be (hopefully) obvious that after the rotation, (1,0) gets mapped to (cos(θ),sin(θ)). this tells you what the first column of the matrix should be (WHY?).

what does (0,1) get mapped to?

(HINT: 0 = cos(π/2), 1 = sin(π/2).

what is cos(π/2 + θ), sin(π/2 + θ)? use the angle-sum identities).
 

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