There is no concept of "rank" in vector space, only concepts of "base" and "dimension". I think the theme must be wrong. It should be that the dimension of vector space is equal to the rank of vector group. The original words are mentioned in the expansion of P 106, the 6th edition of Linear Algebra of Engineering Mathematics (edited by Department of Mathematics of Tongji University). As for why, of course, it is based on the definition.