I am a little bit confused in terms of how these matrices make sense. Why do we always have 1s on the diagonal?

It seems to me that we are trying to get the new coordinates by multiplying the matrix with the original coordinates. However, I don't see how X0 is also at the same position with all these shear matrices...

While doing the quiz, I noticed that when I wrote down the wrong transformations - specifically when I mixed up rotation and scaling in some manner I would get some kind of shearing with my scaled rectangles. During lecture it was mentioned that shearing can be represented as a combination of the basic transformations. I'm wondering if translation is required or can shearing be expressed using just rotation and scaling? If so is there some mathematical justification?

@chenj As with all linear transformations, the origin (in this case, x_0 = [0, 0]) remains in the same location. This is because, regardless of the matrix H, multiplying the matrix with vector [0, 0] produces vector [0, 0]. Also, the shear matrix has 1s on the diagonal by definition, but you can change these values by multiplying a shear matrix with a scale matrix.

@billiam Check out this thread that discusses how to express x-shear matrix in terms of scale and rotation matrices.

Is there a simplier way to write an arbitrary sheer in terms of a rotation and then a single horizontal or vertical sheer?

Try multiplying a rotation matrix and a single horizontal (or vertical) shear matrix to see it's form, and that should give you your answer!