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Math Help - Vector Spaces

  1. #1
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    Vector Spaces

    Which of the following are vector spaces?

    (a) The set of all continuous functions on the interval  [0,1] .

    (b) The set of all non-negative functions on the interval  [0,1] .

    (c) The set of all polynomials of degree exactly  n .

    (d) The set of all symmetric  n \times n matrices, i.e. the set of matrices  A = \{a_{j,k} \}_{j,k = 1}^{n} such that  A^{T} = A .

    So (a) is not a vector space because there is no  \bold{0} vector?

    (b) is a vector space.

    (c) is not a vector space.

    (d) is a vector space.

    Are these correct?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by horacejerry View Post
    Which of the following are vector spaces?

    (a) The set of all continuous functions on the interval  [0,1] .

    (b) The set of all non-negative functions on the interval  [0,1] .

    (c) The set of all polynomials of degree exactly  n .

    (d) The set of all symmetric  n \times n matrices, i.e. the set of matrices  A = \{a_{j,k} \}_{j,k = 1}^{n} such that  A^{T} = A .

    So (a) is not a vector space because there is no  \bold{0} vector?
    so, f(x) = 0 is not a continuous function on [0,1]?

    (b) is a vector space.
    oh, and what would the additive inverse of say, f(x) = 1, be?

    (c) is not a vector space.
    what condition failed?

    (d) is a vector space.
    ok
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    (c) is not a vector space.
    Quote Originally Posted by Jhevon View Post
    what condition failed?
    This vector space isn't closed under addition...

    Ex: \bold u=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0 and \bold v=-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0

    Thus, \bold u\oplus\bold v=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0 +\left(-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0\right) =2a_{n-1}x^{n-1}+...+2a_2x^2+2a_1x+2a_0\notin V

    --Chris
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    This vector space isn't closed under addition...

    Ex: \bold u=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0 and \bold v=-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0

    Thus, \bold u\oplus\bold v=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0 +\left(-a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0\right) =2a_{n-1}x^{n-1}+...+2a_2x^2+2a_1x+2a_0\notin V

    --Chris
    yes, that is one reason why.

    i wanted to know if the poster knew that though
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