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Math Help - Vector Space

  1. #1
    Junior Member pearlyc's Avatar
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    Vector Space

    Are the vectors 6, 3 sin^2x and 2cos^2x in F(-infinity,infinity) the space of all functions from R -> R linearly dependent or linearly independent?
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    Consider: 6 + \left( { - 2} \right)\left[ {3\sin ^2 (x)} \right] + \left( { - 3} \right)\left[ {2\cos ^2 (x)} \right]
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  3. #3
    Junior Member pearlyc's Avatar
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    Thanks for the help, I kinda get the approach but I don't quite know how to present it. Like, where do I start?

    As I know, the way to prove that something is linearly independent or not is through row reduction then compare the ranks. What about this one?
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    QUOTE=pearlyc;132175]As I know, the way to prove that something is linearly independent or not is through row reduction then compare the ranks. What about this one?[/QUOTE]
    You must first learn to general definitions.
    If one can find a nontrivial linear combination that equals the zero then the collection is dependent. Row reductions are applied to matrices and not to general spaces.
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  5. #5
    Junior Member pearlyc's Avatar
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    Is it alright if I were to approach the question like this,

    A linear dependent vector is when we have n vectors and one can be expressed as a linear combination of the rest. When this is impossible, we say the vectors are linearly indepedent.

    Considering,

    a(3sin^2 x) + b(2cos^2 x) = 6
    a(3sin^2 x) + b(2cos^2 x) = 6(sin^2 x + cos^2 x)
    3asin^2 x + 2bcos^2 x = 6sin^2 x + 6cos^2 x

    Comparing coefficients,

    3a = 6
    Therefore, a = 2

    2b = 6
    Therefore, b = 3

    Since the vectors could be expressed as a linear combination, therefore we can say that the vectors are linearly dependent.

    -------------------

    Do you think this working makes sense and would be acceptable?
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    Quote Originally Posted by pearlyc View Post
    Do you think this working makes sense and would be acceptable?
    Yes.
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