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Math Help - need help on dual spaces

  1. #1
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    need help on dual spaces

    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^*.
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  2. #2
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    Quote Originally Posted by Morgan View Post
    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.
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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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