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Math Help - linearly independent problem

  1. #1
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    linearly independent problem

    Let u_{1}, u_{2} be nonzero vectors in R^{n} such that u_{1}u^{T}_{2}=0.

    Show that u_{1}, u_{2} are linearly independent.


    I let u_{1}=(a_{1} a_{2} ... a_{n}) and u_{2}=(b_{1} b_{2} ... b_{n}).

    Then u_{1}u^{T}_{2}=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n  }=0.

    How to continue from here?
    Last edited by deniselim17; March 11th 2010 at 12:05 AM.
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  2. #2
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    Quote Originally Posted by deniselim17 View Post
    Let u_{1}, u_{2} be nonzero vectors in R^{n} such that u_{1}u^{T}_{2}=0.

    Show that u_{1}, u_{2} are linearly independent.


    I let u_{1}=(a_{1} a_{2} ... a_{n}) and u_{2}=(b_{1} b_{2} ... b_{n}).

    Then u_{1}u^{T}_{2}=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n  }=0.

    How to continue from here?


    Suppose u_1,u_2 are lin. depen. \Longleftrightarrow u_1=\lambda u_2\,,\,\,0\neq\lambda\in\mathbb{R} , but then 0=u_1u^T_2=(\lambda u_2)u^T_2=\lambda\left(u_2u^T_2\right)\Longrightar  row u_2=0 , contradiction.

    Tonio
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