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?



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