The image of a tropical linear space

Abstract

Given a tropical linear space $L \subseteq \mathbb{T}^n$ and a matrix $A \in \mathbb{T}^{m \times n}$, the image $AL$ of $L$ under $A$ is typically not a tropical linear space. We introduce a tropical linear space $\mathrm{tropim}_A(L)$, the tropical image, containing $AL$. We show under mild hypotheses that $\mathrm{tropim}_A(L)$ is realizable if $L$ is and apply the tropical image to construct the stable sum of two tropical linear spaces without a disjoint pair of bases.