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Math Help - Vector space Proof

  1. #1
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    Vector space Proof

    Prove that in any set of four vectors in space, at least one of the vectors is a
    linear combination of the other three vectors.

    i got no idea how to approach this question mainly because there are so many scenarios, all 4 in the same plane, 3 in same plane, 2 in same plane and the other 2 in same plane or the remaining 2 in different planes, etc etc. help please
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  2. #2
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    Quote Originally Posted by ah-bee View Post
    Prove that in any set of four vectors in space, at least one of the vectors is a
    linear combination of the other three vectors.

    i got no idea how to approach this question mainly because there are so many scenarios, all 4 in the same plane, 3 in same plane, 2 in same plane and the other 2 in same plane or the remaining 2 in different planes, etc etc. help please
    You obviously mean in the space R^3 .....

    Let u, v, w be independent. Then they form a basis. Which means any other vector will be a linear combination. Details left for you to fill in.
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    Thx for your attempt to help me, but i still don't understand how u prove it though. I know that clearly any 1 of the 4 vectors in the space R^3 can be represented as a linear combination but what i don't get is how u form a "basis" with the independent vectors.
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    Quote Originally Posted by ah-bee View Post
    Thx for your attempt to help me, but i still don't understand how u prove it though. I know that clearly any 1 of the 4 vectors in the space R^3 can be represented as a linear combination but what i don't get is how u form a "basis" with the independent vectors.
    I quote from Basis (linear algebra - Wikipedia, the free encyclopedia):

    "To prove that a set B is a basis for a finite-dimensional vector space V, it is sufficient to show that the number of elements in B equals the dimension of V, and one of the following:
    • B is linearly independent, or
    • span(B) = V."
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  5. #5
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    howdy guys, Ironically i have the exact same question, only problem is I'm a little mathematically challenged... We have not yet been through span's and half the stuff in the wiki page. Can you break it down a little more for me?
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    Do you know what a basis is yet? Do you know that any basis in a vector space will have the same size as any other basis? If so you know {(1,0,0),(0,1,0),(0,0,1)} is a basis for R^3. If there were a vector v that could not be written as a linear combination of these three basis vectors then you could form another set which necessarily is a subset of a basis {(1,0,0),(0,1,0),(0,0,1),v} which has 4 elements which contradicts the result that says every basis has the same size.

    A basis is a spanning set that is linearly independant.
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  7. #7
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    We just went through basis in the lecture today, although it will take me a little time to digest what was said. If you have any more pointers, let me know...

    Cheers
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