Can You Solve This Linear Transformation Equation?

In summary: I was mistaken about the signs in the matrix for R. In summary, the standard matrix for R is [cos\theta -sin\theta] [sin\theta cos\theta]and the inverse of R is [cos\theta sin\theta] [sin\theta -cos\theta]and the standard matrix for F is [1 0] [0 -1]and when multiplied together, the result is not equal to S, which should be [0 1] [1 0].
  • #1
katie_3011
5
0
1. R[tex]\circ[/tex]F[tex]\circ[/tex]R-1=S
where F denotes the reflection in the x-axis
where S is the reflection in the line y=x
where R = R[tex]\pi/4[/tex] : R2 [tex]\rightarrow[/tex] R2

3. An attempt

I have found that the standard matrix for R = [cos[tex]\theta[/tex] sin[tex]\theta[/tex]]
[sin[tex]\theta[/tex] cos[tex]\theta[/tex]]
So therefore, the inverse of R would be the same matrix.

The standard matrix for F = [1 0]
[0 -1]

When I multiplied the matrices together, I got a matrix [1 -1]
[1 1],
which does not equal S, which should be [0 1]
[1 0].

I have tried multiplying out the matrices a few times, and I'm pretty sure this is where my mistake is, but I'm not entirely sure how to multiply cos[tex]\theta[/tex] and sin[tex]\theta[/tex] with actual numbers.

Thanks in advance for your help
 
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  • #2
katie_3011 said:
1. R[tex]\circ[/tex]F[tex]\circ[/tex]R-1=S
where F denotes the reflection in the x-axis
where S is the reflection in the line y=x
where R = R[tex]\pi/4[/tex] : R2 [tex]\rightarrow[/tex] R2

3. An attempt

I have found that the standard matrix for R = [cos[tex]\theta[/tex] sin[tex]\theta[/tex]]
[sin[tex]\theta[/tex] cos[tex]\theta[/tex]]
No, this is not the matrix. To rotate a vector counterclockwise by an angle of theta, the entry in row 1, column 2 should be -sin(theta).
katie_3011 said:
So therefore, the inverse of R would be the same matrix.
Nope, that's not true, either.
katie_3011 said:
The standard matrix for F = [1 0]
[0 -1]

When I multiplied the matrices together, I got a matrix [1 -1]
[1 1],
which does not equal S, which should be [0 1]
[1 0].

I have tried multiplying out the matrices a few times, and I'm pretty sure this is where my mistake is, but I'm not entirely sure how to multiply cos[tex]\theta[/tex] and sin[tex]\theta[/tex] with actual numbers.

Thanks in advance for your help
 
  • #3
katie_3011 said:
1. R[tex]\circ[/tex]F[tex]\circ[/tex]R-1=S
where F denotes the reflection in the x-axis
where S is the reflection in the line y=x
where R = R[tex]\pi/4[/tex] : R2 [tex]\rightarrow[/tex] R2

3. An attempt

I have found that the standard matrix for R = [cos[tex]\theta[/tex] sin[tex]\theta[/tex]]
[sin[tex]\theta[/tex] cos[tex]\theta[/tex]]
So therefore, the inverse of R would be the same matrix.
As Mark44 said, a rotation matrix is anti-symmetric, not symmetric. If the angle is, as here, [itex]\pi/4[/itex] so that [itex]cos(\pi/4)= sin(\pi/4)= \sqrt{2}/2[/itex] and the matrix is
[tex]\begin{bmatrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{bmatrix}[/tex]

Further, the inverse of "rotation by angle [itex]\theta[/itex]" is "rotation through angle [itex]-\theta[/itex]". [itex]cos(-\theta)= cos(\theta)[/itex], [itex]sin(-\theta)= -sin(\theta)[/itex] so changing from [itex]\theta[/itex] to [itex]-\theta[/itex] changes the sign on the "sin" (off-diagonal) but not on the "cos" (diagonal) terms. The matrix rotating by angle [itex]-\theta[/itex] is
[tex]\begin{bmatrix}\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\end{bmatrix}[/tex]

The standard matrix for F = [1 0]
[0 -1]

When I multiplied the matrices together, I got a matrix [1 -1]
[1 1],
which does not equal S, which should be [0 1]
[1 0].

I have tried multiplying out the matrices a few times, and I'm pretty sure this is where my mistake is, but I'm not entirely sure how to multiply cos[tex]\theta[/tex] and sin[tex]\theta[/tex] with actual numbers.

Thanks in advance for your help
 
Last edited by a moderator:
  • #4
I'm pretty sure that the first matrix (the one for R) is correct. These are my assumptions:

If the line for e1 is at an angle theta from the x-axis (assuming that theta is less than pi/4), then the line for e2 would still be in the first quadrant, therefore all of the values would still be positive.

If this is wrong, can you explain to me why?
 
  • #5
katie_3011 said:
I'm pretty sure that the first matrix (the one for R) is correct. These are my assumptions:

If the line for e1 is at an angle theta from the x-axis (assuming that theta is less than pi/4), then the line for e2 would still be in the first quadrant, therefore all of the values would still be positive.

If this is wrong, can you explain to me why?
There was no "e1" or "e2" in what you wrote before so I have no idea what a "line for e2" or "line for e2" would be.
 
  • #6
Both HallsofIvy and I are telling you that your rotation matrix is not correct.
 
  • #7
Thank you, I understand now
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another, while preserving the structures of addition and scalar multiplication.

2. What are some examples of linear transformations?

Examples of linear transformations include rotations, reflections, dilations, and shears. In terms of matrices, any transformation that can be represented by a matrix multiplication is a linear transformation.

3. How do you determine if a transformation is linear?

A transformation is linear if it satisfies two properties: 1) preserving addition, and 2) preserving scalar multiplication. This means that the transformation of the sum of two vectors is equal to the sum of the individual transformations, and the transformation of a scaled vector is equal to the scaled transformation of the original vector.

4. What is the importance of linear transformations?

Linear transformations are used in various fields of science and engineering, including physics, computer graphics, and data analysis. They provide a way to transform and manipulate data in a systematic and efficient manner.

5. Can a linear transformation change the dimension of a vector space?

No, a linear transformation cannot change the dimension of a vector space. The dimension of the output vector space is equal to the number of linearly independent columns in the transformation matrix.

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