Directly Proportional

If the variable y is proportional to x then the formula below, where k is a constant, is true:

y = kx

A constant divided by a constant, is constant, and thus, 1 divided by k is also a constant:

x = \dfrac {1} {k} x

From the above, we can infer that x is directly proportional to y.

Appendix A

Consider the Matrix Multiplication problem below:

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}

The decision to set three of the variabled a,b,c,d to zero and making the fourth nonzero will make either x or y proportional to the fourth, as follows:

x’ = ax

x’ = by

y’ = cx

y’ = dx

This shows how a 2×2 matrix can provide direct proportionality between any new very belt and any old variable. It should make sense that a 3 x 3 matrix would do the same for vectors in XYZ space.