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Thread: Linear map question

  1. #1
    Senior Member
    Joined
    Apr 2010
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    487

    Linear map question

    The question:

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

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

    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:

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

    Any assistance would be great.
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  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
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    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by Glitch View Post
    The question:

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

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

    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:

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

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