need help on dual spaces

Quote:

Originally Posted by Morgan

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

first of all, the characteristic of your base field $F$ must be $\neq 2$ because otherwise $v_1+v_2=v_1-v_2.$ the first part of your question is trivial because $\{v_1+v_2,v_1-v_2 \}$ is a basis for $V$ and so $\beta^*$ is

a basis for $V^*.$ for the second part, we have $2(v_1+v_2)^*(v_1)=(v_1+v_2)^*(v_1+v_2 + v_1 - v_2)=1$ but $2(v_1^* + v_2^*)(v_1)=2.$ so $2(v_1+v_2)^*(v_1) \neq 2(v_1^* + v_2^*)$ and thus $(v_1+v_2)^* \neq v_1^* + v_2^*$ because

$\text{char}(F) \neq 2.$