I feel difficult to understand the dimension of a subset in a vector space, defined in Bernhard Korte and Jens Vygen's Combinatorial Optimization:
We denote by rank($A$) the rank of a matrix $A$. The dimension dim $X$ of a nonempty set $X \subseteq \mathbb{R}_n$ is defined to be $n - \max\{\text{rank}(A): A\text{ is an }n \times n\text{-matrix with }Ax = Ay, \text{ for all }x, y \in X \}.$ A polyhedron $P \subseteq \mathbb{R}_n$ is called full-dimensional if dim $P = n$. Equivalently, a polyhedron is full-dimensional if and only if there is a point in its interior.
I will think $A$ as a linear mapping on $\mathbb{R}_n$. The dimension of a subset $X$ is defined as the nullity of the linear mapping with the maximum rank, over all linear mappings on $\mathbb{R}_n$ such that each maps $X$ to a single point.
Intuitively, I don't understand how the dimension of a subset (or even a polyhedrion) is related to the ranks and nullities of the linear mappings, and neither do I understand the purpose of considering only linear mappings that map $X$ to a single point.
Thanks for any clarification!