My question arises from reading Einstein's The Meaning of Relativity. In particular, on page 12:
If the three components of a vector vanish for one system of Cartesian co-ordinates, they vanish for all systems, because the equations of transformation are homogeneous.
And again on page 13:
In my notation the transformation under discussion is of the form
$$a_{\overline{i}\overline{j}} =e_{\:\overline{i}}^{i}e_{\:\overline{j}}^{j}a_{ij}.$$
This transformation is homogeneous and of the first degree in the $a_{\mu\nu}.$
So, with that added background, here is the original formulation of my question: In Euclidean 3-space, we are given various rectangular Cartesian coordinate system all of the same scale, and related by transformation equations of the form
$$x^{\overline{i}}=a^{\overline{i}}+e_{\:i}^{\overline{i}}x^{i},$$
with the determinant $\left|e_{\:i}^{\overline{i}}\right|=1.$
What does it mean to say that the equations of transformation are homogeneous? In particular what does it mean to say if the components of a vector vanish in any of these coordinate systems, they vanish in all of these coordinate systems, because the equations of transformation are homogeneous?
To me, homogeneous applies to systems of linear equations or differential equations. And the short definition is that the right-hand side is zero.