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Math Help - Vector subspaces

  1. #1
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    Vector subspaces

    I am having difficulty with vector subspaces. It would be helpful if someone could give me an answer to at least one of these two questions.

    1.

    Prove that  W_1= \{(a_1,a_2 \cdot \cdot \cdot ,a_n) \in F^n : a_1+a_2\ + \cdot \cdot \cdot a_n =0 \} is a subspace of F^n, but  W_2= \{(a_1,a_2\,\cdot \cdot \cdot a_n) \in F^n : a_1+a_2 + \cdot \cdot \cdot a_n =1 \} is not


    2.

    Is the set W = { f(x)  \in P(F): f(x)=0 or f(x) has a degree n} a subspace of P(F) if n ≥ 1? Justify your answer
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  2. #2
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     W_1= \{(a_1,a_2,...,a_n) \in F^n : a_1+a_2+...+ a_n =0 \}


    If  a=(a_1,a_2,...,a_n)\in W_1 ,  b=(b_1,b_2,...,b_n) \in W_1 and  t,u \in \mathbb{F} then

    <br />
(ta+ub)_1 + (ta+ub)_2 + ... + (ta+ub)_n = (ta_1+ub_1)+(ta_2+ub_2)+...+ (ta_n+ub_n)

     = t(a_1+a_2+...+a_n) + u(b_1+b_2+...+b_n)

    <br />
= t\cdot0+u\cdot 0<br />

    <br />
= 0 <br />


    Therefore,  ta+ub \in W_1 \implies W1 is a subspace.
    Last edited by abender; January 11th 2010 at 12:04 PM. Reason: forgot a comma
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  3. #3
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    <br />
W_2= \{(a_1,a_2,...,a_n) \in F^n : a_1+a_2 +...+a_n =1 \}<br />


     W_2 is not a vector space because  (0,0,...,0) \not\in W_2. Note that  (0,0,...,0) is simply the zero vector.
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  4. #4
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    Quote Originally Posted by Matharch View Post
    2.[/COLOR]

    Is the set W = { f(x)  \in P(F): f(x)=0 or f(x) has a degree n} a subspace of P(F) if n ≥ 1? Justify your answer[/COLOR]
    [/tex]f(x)= x^2+ 1[/tex] is a polynomial of degree 2. g(x)= -x^2+ 3 is a polynomial of degree 2 also. Is there sum a polynomial of degree 2? Is the set of "polynomials of degree 2" closed under addition?
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  5. #5
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    Thanks guys! It's so simple, I'm almost ashamed that I posted here.
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  6. #6
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    Smile

    Quote Originally Posted by HallsofIvy View Post
    [/tex]f(x)= x^2+ 1[/tex] is a polynomial of degree 2. g(x)= -x^2+ 3 is a polynomial of degree 2 also. Is there sum a polynomial of degree 2? Is the set of "polynomials of degree 2" closed under addition?
    i didn't think that the second question could be answered this way.
    the "Or" made me confused,since it means union.
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