# Thread: Linear Transformations

1. ## Linear Transformations

Hello everyone,

I am currently having trouble with this problem. Any help would be greatly appreciated!

(a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image.

T is the counterclockwise rotation of 45 degrees in $R^2$, v = (2,2)

2. Originally Posted by larson
Hello everyone,

I am currently having trouble with this problem. Any help would be greatly appreciated!

(a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image.

T is the counterclockwise rotation of 45 degrees in $R^2$, v = (2,2)
Notice that $T$ is a linear transformation. You can get the matrix of the transofrmation by figuring out how it acts on its standard basis. Let $\bold{i} = (1,0) \text{ and }\bold{j} = (0,1)$ (just remember to think of them as coloumns, when we deal with $\mathbb{R}^n$ we sometimes things of vectors as an $n\times 1$ matrix). Compute $T(\bold{i})$ and $T(\bold{j})$ (again as coloumns) and form the $2\times 2$ matrix $[ T(\bold{i}) ~ ~ ~ T(\bold{j})]$.

3. Originally Posted by ThePerfectHacker
Notice that $T$ is a linear transformation. You can get the matrix of the transofrmation by figuring out how it acts on its standard basis. Let $\bold{i} = (1,0) \text{ and }\bold{j} = (0,1)$ (just remember to think of them as coloumns, when we deal with $\mathbb{R}^n$ we sometimes things of vectors as an $n\times 1$ matrix). Compute $T(\bold{i})$ and $T(\bold{j})$ (again as coloumns) and form the $2\times 2$ matrix $[ T(\bold{i}) ~ ~ ~ T(\bold{j})]$.
I'm sorry... I'm still kind of confused. How do I find T?

4. ## hi

hi

$\left[\begin{matrix} 1 \\ 0 \end{matrix}\right]$ rotates into $\left[\begin{matrix} cos(\theta) \\ sin(\theta) \end{matrix}\right]$ and $\left[\begin{matrix} 0 \\ 1 \end{matrix}\right]$ rotates into $\left[\begin{matrix} -sin(\theta) \\ cos(\theta) \end{matrix}\right]$

Rotation matrix here will be:
A = $\left[ \begin{matrix} cos(\frac{\pi}{4}) & -sin(\frac{\pi}{4}) \\ sin(\frac{\pi}{4}) & cos(\frac{\pi}{4}) \end{matrix} \right]$

Since $A \left[ \begin{matrix} 1 \\ 0 \end{matrix}\right] = \left[\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}\right]$ and $A \left[ \begin{matrix} 0 \\ 1 \end{matrix}\right] = \left[\begin{matrix} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}\right]$

Now calculate $Av$