1. Problem on Linear Transformations

Find all linear transformations T: R2 --> R2 which carry the line y=x to y=3x.

2. 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.

3. Re: Problem on Linear Transformations

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.

4. 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
$tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}$

5. Re: Problem on Linear Transformations

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.

6. Re: Problem on Linear Transformations

Find all linear transformations T: R2 --> R2 which carry the line y=x to y=3x.
Take a point x,y on l1 and draw a line l through it with slope m. The intersection of l with l2 represents a linear transformation of l1 to l2 for arbitrary m (all linear transformations from l1 to l2). Then solve

T(x,y) = (x’,y’)