In holography, the geometry of gravitational theory should be encoded in its non-gravitational dual. In particular, in gauge/gravity duality, the geometry should be encoded into the color degrees of freedom (matrices) in U(N) supersymmetric Yang-Mills theory. When the Yang-Mills theory is regarded as the low-energy effective theory of a system of N D-branes and open strings between them. Transverse spatial directions emerge from scalar fields, which are N*N matrices with color indices; roughly speaking, the eigenvalues are the locations of D-branes. In the past, it was argued that this simple 'emergent space' picture cannot be used in the context of gauge/gravity duality, because the ground-state wave function delocalizes at large N, leading to a conflict with the locality in the bulk geometry. We show that this conventional wisdom is not correct: the ground-state wave function does not delocalize, and there is no conflict with the locality of the bulk geometry. This conclusion is obtained by clarifying the meaning of the 'diagonalization of a matrix' in Yang-Mills theory, which is not as obvious as one might think. This observation opens up the prospect of characterizing the bulk geometry via the color degrees of freedom in Yang-Mills theory, all the way down to the center of the bulk. [please email a.held@imperial.ac.uk for zoom link or password]