Let $\displaystyle \alpha = \{v_1,v_2\}$ be a basis for $\displaystyle V$ and $\displaystyle \beta^* = \{(v_1 + v_2)^*,(v_1 - v_2)^*\}$ where $\displaystyle v_1^*,v_2^*$ is the canonical basis of the dual. Prove that $\displaystyle \beta^*$ is basis of $\displaystyle V^*$ and $\displaystyle (v_1 + v_2)^*\ne v_1^* + v_2^*.$