Visualizing Linear Transformations

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Click and drag the vectors u (the red vector) and v (the blue vector) to change the matrix of the linear transformation. Note that vector u is the left column of the matrix and v is the right column.

Ctrl + [click and drag] on the background to move the coordinate axes around.
Ctrl + [scroll wheel] to zoom in and out.
What changes when v is held constant and u is moved?
Make a transformation that doubles the area of the picture but keeps the aspect ratio true.
Make a transformation that rotates the picture.
Make a transformation that reflects the picture.
Which configuration of u and v keep the picture "unreflected" and which configurations will reflect it?
What configuration of u and v will make the determinant zero?
How is the sign of the determinant (+ or -) related to the transformation?
How is the magnitude of the determinant related to the transformation?

Marc Renault, Created with GeoGebra