  We show that a smooth linear unsteady velocity field $u(x,t)=A(t)x+f(t)$
solves the incompressible Navier--Stokes equation if and only if the matrix
$A(t)$ has zero trace, and $\dot{{A}}(t)+A^{2}(t)$ is symmetric. In two
dimensions, these constraints imply that $A(t)$ is the sum of an arbitrary
time-dependent traceless symmetric matrix and an arbitrary constant
skew-symmetric matrix. One can, therefore, verify by inspection if an unsteady
spatially linear vector field is a Navier--Stokes solution. In three
dimensions, we obtain a simple ordinary differential equation that $A(t)$ must
solve. Our formulas enable the construction of simple yet unsteady and
dynamically consistent flows for testing numerical schemes and verifying
coherent structure criteria.
