The Buckingham 
theorem states that a dimensionally homogeneous relation among 
physical quantities whose dimensions have rank 
can be rewritten as a relation among 
independent dimensionless products.
To express the result algebraically, let the 
th column of the 
matrix 
contain the exponents of the 
base dimensions in the quantity 
. A monomial
 |
is dimensionless exactly when
 |
Thus the exponent vectors of dimensionless products comprise the null space of 
.
Since its dimension is 
, a basis of that null
space gives 
independent 
groups. The theorem is the central reduction principle of dimensional
analysis.
