  The algebraic $\lambda$-calculus is an extension of the ordinary
$\lambda$-calculus with linear combinations of terms. We establish that two
ordinary $\lambda$-terms are equivalent in the algebraic $\lambda$-calculus iff
they are $\beta$-equal. Although this result was originally stated in the early
2000's (in the setting of Ehrhard and Regnier's differential
$\lambda$-calculus), the previously proposed proofs were wrong: we explain why
previous approaches failed and develop a new proof technique to establish
conservativity.
