Consider the projection operators $P_1, P_2, P_1', P_2'$ with $P_1+P_2=I$, and $\lambda,\mu\in\mathbb C$ with $\lambda\neq\mu$.
Does $\lambda P_1+\mu P_2=\lambda P_1'+\mu P_2'$ imply $P_1'=P_1$ and $P_2'=P_2$?
Does the result change if we instead use real coefficients $\lambda,\mu\in\mathbb R$? Do we need the projectors to be self-adjoint for the result to hold?
A similar problem with projection operators was addressed in this question, but in those cases they didn't have additional constants multiplying the projections.
If you have that $P_1' + P_2' = I$ as well, then we can make the following simplification:
$$(\lambda - \mu)P_1 + \mu P_1 + \mu P_2 = (\lambda - \mu) P_1 + \mu I = (\lambda - \mu)P_1' + \mu I$$
by making the same simplification on the right hand side. Therefore since $\lambda \neq \mu$, we have
$$(\lambda - \mu) P_1 = (\lambda - \mu) P_1' \implies P_1 = P_1'$$