Say I want to construct a flat family of affine schemes using deformation theory. To be more specific, suppose I have a local $k$-algebra $A$, and I want to find a flat family $\mathcal{X} \to T$, where $T$ is $\mathrm{Spec}$ of a ring $T$, and the family has $A$ as central fiber.

To do this, I choose among the space of first-order deformations, $T^1_{A/k}$, some subset $\{t_1, \cdots,t_n\}$ of a basis. This solution can be lifted (modulo obstructions, but that's okay) successively, and if we're lucky, the lifting stops (see for example this article for an algorithm), producing a family $\mathcal{X} \to T$ having $(\langle t_1, \cdots,t_n\rangle/\langle t_1,\cdots,t_n\rangle^2)^\lor$ as tangent space at the origin.

The question is the following: what if I didn't choose the $t_i$ as deformation parameters, but some multiple of them, i.e. $\{c_1t_1,\cdots,c_nt_n\}$, where the $c_i$ are non-zero constants, would I get the same family (up to isomorphism)?

It seems obvious that I should, I even suspect the question doesn't have anything to do with deformation theory, but is purely algebraic - but I haven't been able to see an easy solution.