totally positive Grassmannians

Question

On page 4 of this article https://arxiv.org/pdf/1608.05735.pdf is defined totally positive grassmanian, then he says that if an element $[z] \in {Gr_ {k, m}}$ is defined by a matrix $z$ full range $k \in{m}$ (without loss of generality we can assume that $z$ has real entries) My questions are: Why $z$ is a matrix and why it can assume that $z$ has real entries?