why is translating in 3D space the same as shearing in 4D space?
In a linear transformation system, your origin is always a fixed point, since 0*anything = 0. So imagine you have a cinema screen, and the origin is at the centre of the screen. Using linear transformations, you can rotate, scale or shear the image, what you can't do is move it, since you have a fixed point in the middle.

Now add a dimension, and move your origin out of the screen through this new dimension. Your origin is now at the projector, at &lt;0, 0, 0>, and the centre of your screen is at &lt;0, 0, 1>. So as well as the image transforms you had available before, you can now effectively move your image, by shearing through the space in between the projector and the screen.

Add another dimension and you have the 4D -> 3D case.