# Problem on Linear Transformations

• Apr 22nd 2013, 08:44 AM
Problem on Linear Transformations
Find all linear transformations T: R2 --> R2 which carry the line y=x to y=3x.

• Apr 22nd 2013, 06:55 PM
chiro
Re: Problem on Linear Transformations

Hint: Consider a rotation matrix taking a point on the line y = x to y = 3x by considering rotating both points by the difference of the angle between the two lines.
• Apr 23rd 2013, 05:26 AM
Re: Problem on Linear Transformations
Quote:

Originally Posted by chiro

Hint: Consider a rotation matrix taking a point on the line y = x to y = 3x by considering rotating both points by the difference of the angle between the two lines.

:) yes... I thought about that...but, the problem with that approach is finding the angle between those two lines...i mean, the angle is not something nice. So, I have been looking for an approach in which the matrix of T looks nice.

Anyway, thank you for the reply. Please let me know if you come up with another approach.
• Apr 23rd 2013, 07:57 AM
HallsofIvy
Re: Problem on Linear Transformations
The angle between the two lines is $\theta- \phi$ where $tan(\theta)= 3$ and $tan(\phi)= 1$, the slopes of the two lines. And
$tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}$
• Apr 23rd 2013, 08:36 AM
Re: Problem on Linear Transformations
Quote:

Originally Posted by HallsofIvy
The angle between the two lines is $\theta- \phi$ where $tan(\theta)= 3$ and $tan(\phi)= 1$, the slopes of the two lines. And
$tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}$

:) sometimes we miss the simplest calculations. I calculated the angle in a different way and it didn't look nice. Thanks, now it is done.
• Apr 23rd 2013, 02:16 PM
Hartlw
Re: Problem on Linear Transformations
Quote: