Find three vectors in \(\RR^3\) which are linearly dependent, and are such that any two of them are linearly independent.
Solution
Take \[ v_1=(1,0,0),\qquad v_2=(0,1,0),\qquad v_3=(1,1,0). \] They are linearly dependent because \[ v_1+v_2-v_3=0. \] Any two are linearly independent: \(v_1,v_2\) are standard basis vectors; \(v_3\) is not a scalar multiple of either \(v_1\) or \(v_2\). Hence every pair is independent while the three-vector set is dependent.