I have difficulty understanding the following definition in E.B. Vinberg's algebra book on Chapter 5, Vector Spaces:
Definition 5.2. A basis of a space $V$ agrees with a subspace $U$ if $U$ is a linear span of some basis vectors (i.e., if it is one of the "coordinate subspaces" with respect to this basis).
Isn't it obvious that every basis of a vector space spans a subspace of that vector space? What distinguishes "agrees with a subspace" from "is spanned by the basis of a vector space"?
Thanks, your help is appreciated.