I have read in a book that if $\bf{X} \sim \mathcal{N}(\bf{\mu}, \Sigma)$ and if $\Sigma$ is singular, then possible values of $\bf{X}-\mu$ are constrained to lie in a subspace of $R^d$ with dimension equal to rank($\Sigma$).
What does this mean? Why the dimension of the subspace is lower than the original dimension? How to define this subspace?
If $d=2$ does it mean $X-\mu$ lie its values in $R$?
These seem to relate closely with:
How to find the subspace on which a multivariate normal distribution is concentrated?.