Math subject GRE exam 9768 Q.30 (condition that a basis of a real vector space must satisfy) 


I am sure that A,B and E are wrong, but I do not know which is right C or D, and why, could anyone help me please?  

Take nontrivial scalar multiples of elements of $B $.  This gives a basis  $B'$ which is disjoint from  $B $.  I.e. if $B=\{b_1, b_2,\dots,b_n\}$, let $B'=\{cb_1, cb_2,\dots, cb_n\}$, where $c\neq1$.Then $ B'$ and $B $ are disjoint.    The answer is D.
