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I am trying to find an orthogonal transformation that maps the point (0,5) to the point (3,4).

Now, I found that the transformation matrix M for a reflection in the line y=mx is as follows:

[tex] M = \left(

\begin{array}{cc}

cos(2\theta) & sin(2\theta)\\

sin(2\theta) & -cos(2\theta)

\end{array}

\right)

[/tex]

[tex] \therefore \left(

\begin{array}{cc}

cos(2\theta) & sin(2\theta)\\

sin(2\theta) & -cos(2\theta)

\end{array}

\right)

\left(

\begin{array}{c} 0 \\

5 \\

\end{array}

\right)=

\left(

\begin{array}{c} 3 \\

4\\

\end{array}

\right)

[/tex]

However this means that

[tex] 5sin(2\theta)=3 [/tex]

[tex] -5cos(2\theta)=4 [/tex]

[tex] \frac{5sin2\theta}{-5cos2\theta} = 3/4 [/tex]

[tex] \theta = arctan(-\frac{3}{4}) [/tex]

[tex]\therefore m=\frac{3}{4}

[/tex]

I noticed that if instead I find the angle by taking [tex] 5sin(2\theta)=3 [/tex]

[tex] \theta = 18.43 and tan(\theta)=0.333 [/tex]

Or [tex] -5cos(2\theta)=4 [/tex]

[tex] \theta = 71.56 [/tex] and [tex] tan(71.56)=3[/tex]

Why don't they all yield the same result? isn't this like solving a system of linear equations?

Having said this, I tried to derive the matrix of the transformation myself. I drew the basis vectors i(1,0) and j(0,1) and checked what their new coordinates would be when reflected in a line that makes an angle [tex] \theta [/tex] with the x-axis.

When considering the j(0,1) vector, the angle between j and j' & that between i and i' is [tex] 2\theta [/tex].

The new coordinates for i' would be:

[tex] x = cos(2\theta)

y = sin(2\theta) [/tex]

and those for j' would be:

[tex] x = sin(2\theta)

y = cos(2\theta).

[/tex]

Why would you say that for j' [tex] y=-cos(2\theta)? [/tex]

The only way j' will have negative y coordinates is if the gradient of the line is >45, and if this happens, cos(2x) will be negative (since 90 < 2x < 180, cos(2x) is negative).

Sorry for the very long post, I just wanted to show what I tried before asking any questions. With this being said, could someone please tell me what's wrong with the derivation I attempted? And most of all, why the first one doesn't work?

Thank you!

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# Homework Help: Linear Transformation

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