Theorem: Let V be a vector space over K which has a finite spanning set, and suppose that the vectors $v_1, . . . , v_r$ are linearly independent in V . Then we can extend the sequence to a basis $v_1, . . . , v_n$ of V , where $n ≥ r$.
I dont know what this means, how can a span that is linearly independant be extended to make a basis? Wouldnt that mean adding in some linearly dependant vectors?