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?  

$C$ is also wrong. Basis is a maximal linearly independent set. Thus, $B$ can't be a proper subset of a linearly independent set.
Thus, the correct answer is $D$.
For example, let $V=R^2$ and $B=\{(0,1),(1,0)\}$. Then $\{(1,2),(3,4)\}$ is also a basis of $V$ disjoint from $B$.
