In the culmination of chapter 4, I stated a big idea: any linear transformation in 3D can be specified by just three vectors or nine numbers total. By correctly selecting these nine numbers, we can achieve rotation by any angle about any axis, reflection across any plane, projection onto any plane, scaling by any factor in any direction, or any other 3D linear transformation.
The transformation expressed as “a rotation counterclockwise by 90° about the z-axis” can equivalently be described by what it does to the standard basis vectors e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). Namely, the results are (0, 1, 0), (−1, 0, 0), and (0, 0, 1). Whether we think of this transformation geometrically or as described by these three vectors (or nine numbers), we’re thinking of the same imaginary machine (figure 5.1) that operates on 3D vectors. The implementations might be different, but the machines still produce indistinguishable results.