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Thread: need help on dual spaces

  1. #1
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    need help on dual spaces

    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 $\displaystyle (v_1 + v_2)^*\ne v_1^* + v_2^*.$
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  2. #2
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    Quote Originally Posted by Morgan View Post
    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 $\displaystyle (v_1 + v_2)^*\ne v_1^* + v_2^*.$
    first of all, the characteristic of your base field $\displaystyle F$ must be $\displaystyle \neq 2$ because otherwise $\displaystyle v_1+v_2=v_1-v_2.$ the first part of your question is trivial because $\displaystyle \{v_1+v_2,v_1-v_2 \}$ is a basis for $\displaystyle V$ and so $\displaystyle \beta^*$ is

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

    $\displaystyle \text{char}(F) \neq 2.$
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  3. #3
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    "characteristic?"

    what do you mean by "characteristic?" also, what does mean char(F) and why is not 2?
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