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Math Help - Linear map question

  1. #1
    Senior Member
    Joined
    Apr 2010
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    487

    Linear map question

    The question:

    If {v_1, v_2} are linearly independent in a real vector space V and v_3 = 2v_1 + v_2, is there a linear map T:W -> \mathbb{R}^2 where W = span(v_1, v_2) such that

    T(v_1) = \[<br />
\left( {\begin{array}{c}<br />
 1 \\<br />
 2 \\<br />
 \end{array} } \right)<br />
\], T(v_2) = \[<br />
\left( {\begin{array}{c}<br />
 -3 \\<br />
 2 \\<br />
 \end{array} } \right)<br />
\], T(v_3) = \[<br />
\left( {\begin{array}{c}<br />
 -1 \\<br />
 3 \\<br />
 \end{array} } \right)<br />
\]?

    I'm not sure how to do this. I'm thinking it has something to do with the fact that a linear map needs to satisfy:

    T(\lambda v_1 + ... + \lambda_n v_n) = \lambda_1 T(v_1) + ... + \lambda_n T(v_n)

    Any assistance would be great.
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  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Glitch View Post
    The question:

    If {v_1, v_2} are linearly independent in a real vector space V and v_3 = 2v_1 + v_2, is there a linear map T:W -> \mathbb{R}^2 where W = span(v_1, v_2) such that

    T(v_1) = \[<br />
\left( {\begin{array}{c}<br />
 1 \\<br />
 2 \\<br />
 \end{array} } \right)<br />
\], T(v_2) = \[<br />
\left( {\begin{array}{c}<br />
 -3 \\<br />
 2 \\<br />
 \end{array} } \right)<br />
\], T(v_3) = \[<br />
\left( {\begin{array}{c}<br />
 -1 \\<br />
 3 \\<br />
 \end{array} } \right)<br />
\]?

    I'm not sure how to do this. I'm thinking it has something to do with the fact that a linear map needs to satisfy:

    T(\lambda v_1 + ... + \lambda_n v_n) = \lambda_1 T(v_1) + ... + \lambda_n T(v_n)

    Any assistance would be great.
    Is T\left(2v_1+v_2\right)=2T(v_1)+T(v_2) equal to T(v_3)?
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  3. #3
    Senior Member
    Joined
    Apr 2010
    Posts
    487
    Nope. Thanks.
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