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Math Help - Linear Algebra: Vector Space!

  1. #1
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    Linear Algebra: Vector Space!

    Hello,

    I need help in this linear Algebra problem...

    Prove that W1 = {(a1, a2, a3....a3) E F^n: a1+a2+a3+...+an = 0} is a subspace of F^n, but
    W2 = {(a1, a2, a3....a3) E F^n: a1+a2+a3+...+an = 0} is not...



    Thanks for the help in advance!
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  2. #2
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    What is written leads to W_1=W_2, perhaps we should replace = by \neq in the definition of W_2. If so, since a subspace of a vector space E must contain 0_{E} and that 0+...+0=0, we can conclude that W_2 doesn't contain (0,...,0) and isn't a vector space.

    By the way, W_1 contains (0,...,0). So the only thing we have to prove now is that for every \lambda ,\mu\in k (if F is a k-vector space), (a_1,...,a_n),(b_1,...,b_n)\in F^n\Rightarrow \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)\in F^n.

    Let (a_1,...a_n) and (b_1,...,b_n) be two elements of F^n, since \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)=(\lambda a_1+\mu a_1,...,\lambda a_n+\mu a_n),

    \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)\in F^n
    \Leftrightarrow \lambda a_1+\mu b_1+...+\lambda a_n+\mu b_n)=0
    \Leftrightarrow \lambda(\underbrace{a_1+...+a_n}_{=0})+\mu(\underb  race{b_1+...+b_n}_{=0})=0
    \Leftrightarrow \lambda .0+\mu .0=0
    \Leftrightarrow 0=0

    Therefore \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)\in F^n, and W_1 is a subspace of F^n
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