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Math Help - How do I determine which space these vectors span?

  1. #1
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    How do I determine which space these vectors span?

    Hi guys. I'm gonna right down two different problems. The first one I understand but not the second. Find the space the vectors span.

    1. a=(3,5,0) b=(1,2,0) sp(a,b)= (3\alpha+\beta, 5\alpha+2\beta, 0) \rightarrow sp(a,b) = (\lambda_1, \lambda_2, 0)

    2. c=(1,1,2) d=(1,0,1) sp(c,d) = (\alpha+\beta, \alpha, 2\alpha+\beta) \rightarrow sp(c,d) = ???

    How do I continue to get an answer like the first question - something of the form (\lambda_1, 0, \lambda_2)...(I know that's not correct, just giving an example)
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  2. #2
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    Quote Originally Posted by jayshizwiz View Post
    Hi guys. I'm gonna right down two different problems. The first one I understand but not the second. Find the space the vectors span.

    1. a=(3,5,0) b=(1,2,0) sp(a,b)= (3\alpha+\beta, 5\alpha+2\beta, 0) \rightarrow sp(a,b) = (\lambda_1, \lambda_2, 0)

    2. c=(1,1,2) d=(1,0,1) sp(c,d) = (\alpha+\beta, \alpha, 2\alpha+\beta) \rightarrow sp(c,d) = ???

    How do I continue to get an answer like the first question - something of the form (\lambda_1, 0, \lambda_2)...(I know that's not correct, just giving an example)
    There is no "something of the form" in case (2), but note that if we write (x,y,z)=(\alpha+\beta, \alpha, 2\alpha+\beta) , then

    it is true that x+y-z=0 , and this last is the equation of a plane in the 3-dimensional space, so

    Span\{c,d\}=\{(x,y,z)\,;\,x+y-z=0\}

    By the way, also (1) can be written in this thrifty way: span\{a,b\}=\{(x,y,z)\,;\,z=0\}= the xy-plane in the 3-dim. space.

    Tonio
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