  We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear
transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild
conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1
A+\mathcal{L}_2 A|\geq
\left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mathcal{L}_2)|^{1/d}\right)^d|A|-
o(|A|).$$ This result corrects and confirms the two-summand case of a
conjecture of Bukh and is best possible up to the lower-order term for many
choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. As an application, we prove a
lower bound for $|A + \lambda \cdot A|$ when $A$ is a finite set of real
numbers and $\lambda$ is an algebraic number. In particular, when $\lambda$ is
of the form $(p/q)^{1/d}$ for some $p, q, d \in \mathbb{N}$, each taken as
small as possible for such a representation, we show that $$|A + \lambda \cdot
A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to
the lower-order term and extends a recent result of Krachun and Petrov which
treated the case $\lambda = \sqrt{2}$.
