# Surjectivity of a linear mapping from positive real vectors to real vectors

Consider a linear function $$A: \mathbb{R}^m \rightarrow \mathbb{R}^n$$. To check for surjectivity of $$A$$, we can simply compute its rank; if it is equal to $$n$$, this mapping is surjective. But what if we restrict the definition space of $$A$$ to $$\mathbb{R}^m_+ := \{ x\in \mathbb{R}^m: \; x_j \geq 0 \; \forall j \in \{1,\dots , m\} \}$$?

For example, consider the identity matrix. Obviously, it is surjective defined on $$\mathbb{R}^m$$, but not on $$\mathbb{R}^m_+$$.

Is there a helpful criterion to check for surjectivity in such a setting?

Of course, there are some settings where you can prove something like that. For instance, if $$A$$ has full rank (we always assume $$m>n$$), and wlog the first $$n$$ rows of A are linearly independent, and if the $$n+1$$th row $$A[n+1]= - \sum_{j=1}^n A[j]$$ , then we should be able to prove surjectivity of $$A$$ on $$\mathbb{R}^m$$ I think. But can we derive a more general criterion?