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Math Help - Span help

  1. #1
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    Span help

    This is NOT for a graded paper, as my other thread was closed. I have a test on Friday, and I'm trying to understand spans. I just attached it because I thought it would be better formatted.

    Here are the typed questions instead:

    Show that the vectors (1,2-1),(3,1,1),(1,1,0),(1,-3,3) are linearly dependent by writing the zero vector as a nontrivial linear combination of them.

    When is the vector (x,y,z) in the span of {(1,2-1),(3,1,1)(1,-3,3)}?

    Hope that's better, as I still need some help.

    Thanks
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by shibble View Post
    This is NOT for a graded paper, as my other thread was closed. I have a test on Friday, and I'm trying to understand spans. I just attached it because I thought it would be better formatted.

    Here are the typed questions instead:

    Show that the vectors (1,2-1),(3,1,1),(1,1,0),(1,-3,3) are linearly dependent by writing the zero vector as a nontrivial linear combination of them.

    When is the vector (x,y,z) in the span of {(1,2-1),(3,1,1)(1,-3,3)}?
    If and only if there are scalars a,b,c such that (x,y,z)=a*(1,2,-1)+b*(3,1,1)+c*(1,-3,3). Hence just solve this system of linear equations for the unknowns a,b,c for (x,y,z)=(0,0,0).
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  3. #3
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    Quote Originally Posted by Failure View Post
    If and only if there are scalars a,b,c such that (x,y,z)=a*(1,2,-1)+b*(3,1,1)+c*(1,-3,3). Hence just solve this system of linear equations for the unknowns a,b,c for (x,y,z)=(0,0,0).
    Is that just for the second part or the first part as well?
    For example, for the first one would I just do
    a 3b c d 0
    2a b c -3d 0
    -a b -3d 0

    and just solve for this?
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by shibble View Post
    Is that just for the second part or the first part as well?
    For example, for the first one would I just do
    a 3b c d 0
    2a b c -3d 0
    -a b -3d 0

    and just solve for this?
    That's for the first question. And I hope you understand that there always exists the so called trivial solution, that is a=b=c=0. The given vectors are linearly-dependent if and only if there also exist non-trivial solutions of this system of linear equations.

    I have already given the answer to the second question.
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