# Thread: Linear Transformations

1. ## Linear Transformations

Hi,
I have a problem regarding linear transformations.
Consider the triangle in R^2 with vertices (1,1), (3,2) and (2,3). Find a linear transformation T:R^2 to R^2 which firstly doubles the vertical distance from the origin to each of the vertices, then rotates the triangle through an angle of 5pi/6 anticlockwise about the origin, and finally reflects it across the line y=x.
Thankyou in advance for any assistance provided.

2. Do it one step at a time. Any linear transformation on $R^2$ can be represented as a 2 by 2 matrix. And you want
$\begin{bmatrix} a & b \\ c & d\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix}x \\ 2y\end{bmatrix}$ Do that multiplication on the left, set components equal to those on the right and you can see what a and b must b.

You could do the same thing for the rotation but it is simpler to use the fact that a counter clockwise rotation, about the origin, through $\theta$ is represented by the matrix
$\begin{bmatrix}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{bmatrix}$

Doing both is equivalent to multipying the matrices. Be sure you multiply in the right order: the matrix applied first on the right.

3. Thankyou for your assistance halls of ivy
I understand the technique you suggested to complete the question, its just that i'm not sure where to get started. Specifically what do i need to do given the vertices of this triangle, in order to get to the stage where i can actually start to use your technique and perform the linear transformations using matrix operations?