If we rescale the i th fundamental unit by a factor of α α i , then q j gets rescaled by α α 1 − − m 1 j α α 2 − − m 2 j ⋯ ⋯ α α ℓ ℓ − − m ℓ ℓ j , where m i j is the ( i , j ) th entry of the dimensional matrix. In order to convert this into a linear algebra problem, we take logarithms (the base is irrelevant), yielding [ log ⁡ ⁡ q 1 ⋮ ⋮ log ⁡ ⁡ q n ] ↦ ↦ [ log ⁡ ⁡ q 1 ⋮ ⋮ log ⁡ ⁡ q n ] − − M T [ log ⁡ ⁡ α α 1 ⋮ ⋮ log ⁡ ⁡ α α ℓ ℓ ] , which is an action of R ℓ ℓ on R n . We define a physical law to be an arbitrary function f : : ( R + ) n → → R such that ( q 1 , q 2 , … … , q n ) is a permissible set of values for the physical system when f ( q 1 , q 2 , … … , q n ) = 0 . We further require f to be invariant under this action. Hence it descends to a function F : : R n / im ⁡ ⁡ M T → → R . All that remains is to exhibit an isomorphism between R n / im ⁡ ⁡ M T and R p , the (log) space of pi groups ( log ⁡ ⁡ π π 1 , log ⁡ ⁡ π π 2 , … … , log ⁡ ⁡ π π p ) .