Adjust sliders a, b, and c to change the matrix M. This changes the function F=xT*M*x so the constant surfaces of F (ellipses or hyperbolas) are modified in the view.
Move the blue point at the tip of the red arrow to change the x vector of the equation Mx=b. The blue vector b is the result of the transform. The dashed blue vector, is the vector b moved to the tip of the vector x. Note that this vector is normal to the constant surfaces of the function F, and is actually equivalent to half the gradient of the function F.
The purple arrow is resticted to the constant surface F=1. Move the green point at the tip of the purple arrow to change the position of this vector. The green arrow is the result of M times the purple arrow. A right triangle symbol is shown in order to indicate that the resulting green vector is always normal to the constant surface F=1 since it is 1/2 the gradient of F.