We explain how to define, in matrix (gauge) field theory, the notion of the effective action of k ``probe'' branes in the presence of N ``background'' branes on which the field theory lives. The analysis of the large N planar diagram expansion which computes the effective action yields a simple and generic mechanism explaining the emergence of holographic space dimensions. The resulting effective action matches the non-abelian D-brane action in the closed string background dual to the field theory. We shall discuss briefly a few explicit examples, and also provide a general introduction to the notion of emerging space.