We follow a common thread to express linear transformations of vectors and bivectors from different fields of physics in a unified way. The tensorial representations are coordinate independent and assume a compact form using Clifford products. As specific examples, we present (a) the inertia tensor as a vector-to-vector as well as a bivector-to-bivector linear transformation; (b) the Newtonian tidal acceleration; and (c) the Riemann tensor corresponding to a Schwarzschild black hole as a bivector-to-bivector tensorial transformation. The resulting expressions have a remarkable similarity when expressed in terms of geometric products.