Prove or give a counterexample If v1 v2 v3 v4 is a basis of

Prove or give a counterexample: If v_1, v_2, v_3, v_4 is a basis of V and U is a subspace of V such that v_1, v_2 epsilon U and v_3 epsilon U and v_4 epsilon U, then v_1, v_2 is a basis of U.

Solution

False

Consider

v1=(1,0,0,0),v2=(0,1,0,0),v3=(0,0,1,0) and v4=(0,0,0,1) as basis for V

Let U be a subspace of V spanned by

(1,0,0,0),(0,1,0,0),(0,0,1,1)

U contains v1,v2 but v1,v2 is not a basis for U