To demonstrate systematic, linear algebra-based, dimensional analysis to derive a scaling relationship among the design parameters of MRI gradient and harmonic shim coils.The dimensions of five physical quantities relevant for gradient coil design (inductance, gradient amplitude, inner diameter [Only when the unit of turns was included did the linear algebra-based analysis uniquely produce the known scaling relationship that gradient inductance is proportional to gradient efficiency squared timesSystematic application of linear algebra-based dimensional analysis can provide new insight in gradient and shim coil design by revealing fundamental scaling relations and helping to guide the design and comparison of coils with different diameters.