Matrix Problem

[plasmatic]

[solved]]

 

[Ishuda]

Hi, how would one approach proving this?

\(\displaystyle \mbox{Let }\, l_{\theta}\, \mbox{ be line in }\, \mathbb{R}^2\, \mbox{ going through }\, (0,\, 0)\, \mbox{ which makes}\)

\(\displaystyle \mbox{an angle }\, \theta\, \mbox{ above the positive }\, x\mbox{-axis. Consider the linear}\)

\(\displaystyle \mbox{transformation associated to the matrix:}\)

. . . . .\(\displaystyle \pi_{\theta}\, =\, \left(\, \begin{array}{cc}\cos^2(\theta)&\cos(\theta)\sin(\theta) \\ \cos(\theta)\sin(\theta)&\sin^2(\theta)\end{array}\,\right)\)

\(\displaystyle \mbox{For every point }\, v\, \in\, \mathbb{R}^2,\, \mbox{ show that }\, \pi_{\theta}(v)\, \mbox{ lies on the line }\, l_{\theta}.\)

My current understanding is that L theta is just the rotation transformation? Which is:

. . . . .\(\displaystyle R_{\alpha}\, =\, \left[\, \begin{array}{cc}\cos(\alpha)&-\sin(\alpha) \\ \sin(\alpha)&\cos(\alpha) \end{array}\, \right]\)

and then you apply a generic point (x,y) to both of these tranformations and prove they're equal? But I'm having trouble doing that.

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

HINT:
What is the equation of the line \(\displaystyle \scriptstyle{l}_{\theta}\)?

Choose an arbitrary point (x,y) and apply the transformation
\(\displaystyle \begin{pmatrix}w\\z\end{pmatrix}\, =\, \pi_{\theta}(v)\, =\, \pi_{\theta}\, \begin{pmatrix}x\\y\end{pmatrix}\)
What is the relationship between w and z?

 

[Ishuda]

My current working is: http://puu.sh/mYhE5/e9a25b2da0.jpg and then I get stuck. I think I answer your two questions in my working, although I don't introduce a w and z.

Ah, I see. One mistake is in assuming a rotation must occur. The only two reasons for defining the line \(\displaystyle \scriptstyle{l}_{\theta}\) is
(1) Identifying a line in the plane, i.e. the is
\(\displaystyle \scriptstyle{l}_{\theta}\): y = tan(\(\displaystyle \theta\)) x
and
(2) Defining the angle \(\displaystyle \theta\) for the transformation matrix \(\displaystyle \pi_{\theta}\).

So, labeling the points for the work you showed, you have
\(\displaystyle \begin{pmatrix}w\\z\end{pmatrix}\, =\, \pi_{\theta}(v)\, =\, \pi_{\theta}\, \begin{pmatrix}x\\y\end{pmatrix}\)
with
w = x cos²(\(\displaystyle \theta\)) + y sin(\(\displaystyle \theta\)) cos(\(\displaystyle \theta\))
z = y sin(\(\displaystyle \theta\)) cos(\(\displaystyle \theta\)) + x sin²(\(\displaystyle \theta\))

the equation for w is correct but the equation for z is incorrect. Fix the equation for z and see if you can get a relationship between w and z.