Select one of the six pre-defined linear transformation types and enter an "amount" value to define a linear transformation. For example, if you select the rotation transformation, then "amount" represents the rotation angle (measured in radians counterclockwise from the positive x-axis), while if you select the horizontal stretch, then "amount" represents the horizontal scaling factor. In particular, a horizontal stretch with an "amount" of -1 will simply be the reflection across the y-axis, while an "amount" of 0 will yield the projection onto the y-axis.[br][br]If you select the "Custom" transformation instead, then you will see the vectors [math]\vec{a}[/math] and [math]\vec{b}[/math] that you can drag to define the outputs for the two standard basis vectors [math]\left[1,0\right][/math] and [math]\left[0,1\right][/math], respectively. This means that the custom transformation is simply the matrix transformation corresponding to the 2x2 matrix whose columns are [math]\vec{a}[/math] and [math]\vec{b}[/math].[br][br]Once you have defined your transformation, if you have the "Vector" checkbox ticked, then the plot will show you an input vector [math]\text{\vec{v}}[/math] and its corresponding output vector [math]T\left(\vec{v}\right)[/math]. You can move [math]\text{\vec{v}}[/math] by dragging its terminal point and then observe how [math]\text{T(\vec{v})}[/math] changes.[br][br]If you tick the "sum" or "multiple" checkboxes, you can verify that each linear transformation preserves vector addition and scalar multiplication, i.e. [math]T\left(\vec{u}+\vec{v}\right)=T\left(\vec{u}\right)+T\left(\vec{v}\right)[/math] and [math]T\left(c\vec{v}\right)=cT\left(\vec{v}\right)[/math]. The vectors [math]\vec{u}[/math], [math]\vec{v}[/math], and [math]c\vec{v}[/math] can all be moved by dragging their terminal points.[br][br]If you tick the "Image" checkbox, then you can see how the selected linear transformation affects an image of a cat. Each point in the original image corresponds to a vector (the position vector for that point), and transforming that vector reveals the location of the corresponding point in the transformed image.