Linear Transformation about the x-axis

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Homework Help Overview

The discussion revolves around finding a linear transformation from R3 to R3 that rotates an object clockwise by an angle θ around the x-axis. The original poster presents a matrix they believe represents this transformation but expresses uncertainty about how to demonstrate its derivation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the derivation of the transformation matrix and the properties it must satisfy. Questions arise regarding the necessity of showing the transformation's derivation and whether to consider homogeneous equations. There is also a suggestion to analyze how the basis vectors transform under the rotation.

Discussion Status

The discussion is active, with participants exploring different aspects of the problem. Some guidance has been offered regarding the transformation of basis vectors, and there seems to be a general understanding that the matrix should reflect the correct transformations of these vectors. However, there is no explicit consensus on the steps required for the original poster to demonstrate their solution.

Contextual Notes

The original poster expresses confusion despite reviewing lecture notes and books, indicating a potential gap in understanding the underlying concepts of linear transformations and their properties.

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Homework Statement



Find a linear transformation T from R3 to R3 which has the effect of rotating an object clockwise by angle θ around the x-axis.

Homework Equations



none

The Attempt at a Solution



I know that I should work with matrices to show how I came up to the final matrix, which i think has to be the following:

|x'| | 1 0 0 |
|y'| = | 0 cosθ -sinθ|
|z'| | 0 sinθ cosθ|

However, I don't know how to proceed to show how I came to the following matrix.

Any help is appreciated..
 
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Well, how did you come up with that? Don't you remember what you did? In any case, the problem doesn't really ask you to show how you arrived at that. What you do need to do is show that the matrix has the required properties.
 
Well that is just the solution I have found.

What do you mean that the matrix has the required properties?

Do I need to figure out the homogenous equations first?

I am a bit confused with all these, I've been reading lecture notes and books for ages but none of them seems to make sense to me.
 
Figure out how (1,0,0), (0,1,0) and (0,0,1) should transform under your rotation. The resulting vectors are the columns of your matrix. Isn't that what you did? Then you are done.
 
So there are no further steps needed to show?

That's what I did actually...
 
Not that I can think of. The three basis vectors transform to the correct place. Everything else will just follow.
 

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