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Math Help - Vectors Question

  1. #1
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    Vectors Question

    I don't know why vectors {v1, v2, v3} form a basis for R3. Please explain.
    Thank you
    Attached Thumbnails Attached Thumbnails Vectors Question-mathsquestion-orthogonal.jpg  
    Last edited by al1850; March 16th 2008 at 10:11 PM.
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  2. #2
    Moo
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    Hello,

    It's quite strange, because v1=-v2, so (v1,v2,v3) can't be a basis of R3
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    Quote Originally Posted by al1850 View Post
    I don't know why vectors {v1, v2, v3} form a basis for R3. Please explain.
    Thank you
    As implied, there's probably a typo in the question.

    Note: If a set of vectors spans and is linearly independent, then it's a basis.
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    Moo
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    Or "if a set of vectors is linearly independent and its dimension is the dimension of R3 for example, then it's a basis"
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    Don't think there is a typo in the question
    'Cos that's exactly how it is asked in a Maths assignment.
    Regardin the question, is that because V2=-V1, thus V1 and V2 are linearly dependant.
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    Quote Originally Posted by al1850 View Post
    Don't think there is a typo in the question
    'Cos that's exactly how it is asked in a Maths assignment.
    Regardin the question, is that because V2=-V1, thus V1 and V2 are linearly dependant.
    1. v1 and v2 are independent and therefore by definition cannot be part of a basis. At best, the set of vectors span. But .....

    2. The set of vectors do not span. eg. You cannot construct any vector of the form (a, 0, 0) from them.
    Alternatively, the dimension of the set of vectors is 2 (since only two of them are independent). But the dimension of R3 is ...... well, 3.

    There has to be a typo. I suggest you discuss this with your professor.
    Last edited by mr fantastic; March 16th 2008 at 05:40 PM.
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  7. #7
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    finally, got it corrected, the -1/√2 in second column second row should be 1/√2
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  8. #8
    Moo
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    Ok, this seems better ^^

    Now, prove that if there are a,b,c such as av1 + bv2 + cv3 = 0 thus a = b = c = 0
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  9. #9
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    Quote Originally Posted by Moo View Post
    Ok, this seems better ^^

    Now, prove that if there are a,b,c such as av1 + bv2 + cv3 = 0 thus a = b = c = 0
    Another way might be to see how the standard basis (1, 0, 0), (0, 1, 0), (0, 0, 1) could be obtained from v1, v2, v3 .....

    For starters, v1 + v2 = ...... therefore .....
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