# Rotation and shear mapping of Linear algebra

I created simple examples of the rotation and shear mapping of linear algebra for understanding.

My examples this repository.

This visualized example was very helpful for understanding eigenvectors and eigenvalues.

# Rotation

The rotation uses below rotation matrix.
$A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

A rotated vector is represented like this;

$rotatedVector = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} v$

E.g.

$\begin{bmatrix} \cos90 & -\sin90 \\ \sin90 & \cos90 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$

## Example

This is my example code.

Origin image here.

Rotate it 90 degrees.

# Shear mapping

The shear mapping uses below matrix.
$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$

A sheer mapping vector is represented like this;

$shearedVector = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} v$

E.g.

$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

The x-axis of the vecter is increased, but the y-axis is not changed. Any other vectors are the same behaviou​r​, only the x-a​xis values are changed.

# Example

This is my example code.

Sheard image.