Find the Reflection Line of a Matrix, and Analyze the Transformation

[Ascendant0]

Homework Statement

Solve the matrix to find the reflection line. Then, analyze the transformation

Relevant Equations

$$ \begin{bmatrix} -1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2 & 1/2 \end{bmatrix} $$

I just want to make sure I'm doing this right. I know how to do the rotation, but reflection isn't demonstrated in the text. From what I'm seeing in the book, it seems like you take the matrix and simply multiply it by $\begin{bmatrix} x \\ y \end{bmatrix}$

Assuming that's the case, I get
$-x/2 - \sqrt{3}y/2$ as well as
$-\sqrt{3}x/2 + y/2$

Hopefully this is right so far. If not, where am I going wrong? If so, what do I do with this from here? I'm not sure how to turn this into one equation for the reflection line?

 

[Filip Larsen]

What do you know will always hold true (by definition) when transforming points that are on the reflection line?

 

[WWGD]

What do you know will always hold true (by definition) when transforming points that are on the reflection line?

And add to Filip's comment by using eigenvalues.

 

[Ascendant0]

What do you know will always hold true (by definition) when transforming points that are on the reflection line?

That the transformation won't change them.

I was actually just coming back here to post before I saw your reply, as I think I get it now. I believe I should be able to take either equation, set it to 0, solve for y, and that should give me the right answer? Only thing I'm not sure about is one gives $y = -x \sqrt(3)$ while the other gives $y = -x/ \sqrt(3)$?

 

[Filip Larsen]

That the transformation won't change them.

Not quite, but you know a general point on the line must transform to another point on the line. There is however one special point that always transforms to itself no matter what the elements of you transform are, so that means should a reflection line exist it must also go through this point which can be helpful when you have to model what it means for a point to lie on the line.

I was actually just coming back here to post before I saw your reply, as I think I get it now. I believe I should be able to take either equation, set it to 0, solve for y, and that should give me the right answer? Only thing I'm not sure about is one gives $y = -x \sqrt(3)$ while the other gives $y = -x/ \sqrt(3)$?

Equating either transformed coordinate with zero will "just" establish an equation for points that are transformed to either the y- or x-axis, which is not what you seek. If you have a random point (x,y) how do you determine if it lies on a line?

 

[Filip Larsen]

And add to Filip's comment by using eigenvalues.

But general points on a reflection line (if it exists) for a general transform will not map to themselves. Granted, in this specific case the transform has determinant -1 so it can be modeled as a rotation and an axis flip, but not sure trying to formulate this as an eigenvalue problem is useful for the level of the OP.

 

[Filip Larsen]

I know how to do the rotation, but reflection isn't demonstrated in the text.

If you are comfortable with rotations then observing your transform has determinant -1 perhaps you can (as mentioned) try model the the transform as a rotation composed with an axis flip (the 2x2 matrix for this is very simple). Geometrically this means the reflection line will be the points that when rotated and then having one coordinate negated will end up on same point (same in this case because determinant -1 preserves lengths). This also means there is a very simple geometric relationship between the reflection line and the rotation angle and selected flip-axis.

 

[Ascendant0]

Not quite, but you know a general point on the line must transform to another point on the line. There is however one special point that always transforms to itself no matter what the elements of you transform are, so that means should a reflection line exist it must also go through this point which can be helpful when you have to model what it means for a point to lie on the line.


Equating either transformed coordinate with zero will "just" establish an equation for points that are transformed to either the y- or x-axis, which is not what you seek. If you have a random point (x,y) how do you determine if it lies on a line?

Well, your first statement here is completely throwing me off. If it's a line of reflection, and a point is right on that line, how is it still changed? Isn't a line of reflection the line which it's reflected across, so if it's on that line, then there's nothing to reflect across, as it's already on the line it would be reflected across?

I'm seeing that analogous to something like "I want you to move an object the same distance away from this line on the opposite side of the line" then giving me something directly on the line to move it across. Well, if it's on it, how is it going anywhere?

I think I'm looking at this wrong, and then the book is giving us $sin\theta$ and $cos\theta$, but I have no clue what we would even plug in for $\theta$ considering we aren't given any angles, just a 2x2 matrix. The three people I talk to from class aren't getting this problem either, our book is absolutely horrible at explaining (or I should say *not* explaining), and our homework is due tomorrow before he has his office hours

Anyway, to add to your question at the end there, to find if a point is on a line, I would plug those values into the slope equation and make sure I get the correct answer. If I do, it falls on that line; if not, then it isn't on the line. I don't see how to apply that to the 2x2 matrix though?

 

[Ascendant0]

If you are comfortable with rotations then observing your transform has determinant -1 perhaps you can (as mentioned) try model the the transform as a rotation composed with an axis flip (the 2x2 matrix for this is very simple). Geometrically this means the reflection line will be the points that when rotated and then having one coordinate negated will end up on same point (same in this case because determinant -1 preserves lengths). This also means there is a very simple geometric relationship between the reflection line and the rotation angle and selected flip-axis.

So with a rotation, the way it's explained in the book is you take the matrix and multiply it by the column matrix $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$, so we basically just take the first value in each equation, and the first is for the x axis, the second is for the y axis, like this:

$\begin{bmatrix} -1/2 & \sqrt(3)/2 \\ -\sqrt(3)/2 & -1/2 \end{bmatrix} \begin{bmatrix} 1 \\0 \end{bmatrix}$, = $-(i)1/2 +(j)\sqrt(3)/2$

No explanation why it works like that, nor any explanation on the method you'd use for a reflection. So, all I know is that's what they do in the textbook. They multiply the 2x2 by that column matrix, which essentially just takes the first column of the original matrix, and the first value they define as i, the second as j. But again, as far as exactly why they do it that way, and what it has to do with the sin an cos matrices they mention about these transformations, they don't make any sense of that in the text

 

[Ascendant0]

Equating either transformed coordinate with zero will "just" establish an equation for points that are transformed to either the y- or x-axis, which is not what you seek. If you have a random point (x,y) how do you determine if it lies on a line?

Ok, so I attached this here, as they cover the same problem again later on in the next chapter. According to this here, it states the answer I wrote earlier $y = -x\sqrt(3)$ is the reflection line, but you said it wasn't???

From what they further discuss in this part, it seems like for the equation of the line, you just solve for y, so basically multiply the matrix M by $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ , which is essentially giving you the y value of the problem.

For their final solution, it seems like you do set it to 0 to put it in $y = mx + b$ form. I mean it's what I did, it's what they state later in the book here, so what am I missing here?

 

[Filip Larsen]

Ok, so I attached this here, as they cover the same problem again later on in the next chapter. According to this here, it states the answer I wrote earlier $y = -x\sqrt(3)$ is the reflection line, but you said it wasn't???

In your first post and later you said you multiplied the transformation with (x,y) and then equated to this to zero which then gives the (wrong) result $y = -x/\sqrt{3}$. Since you have the answer readily available in your textbook you should be able to spot the difference between your and their answer.

On top of that I must admit that I probably did add confusion more than help by initially replying about what a reflection line means for a general transform and not the specific one you posted. So just ignore that and focus on what your textbook does to get to the correct result.

For their final solution, it seems like you do set it to 0 to put it in $y = mx + b$ form. I mean it's what I did, it's what they state later in the book here, so what am I missing here?

For what I have read so far, I would guess you are missing that you need to establish an equation like the one you quoted from your textbook in post #10. Are you good with why they equate the transformed point to the point itself?