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Thread: linearly independent problem

  1. #1
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    linearly independent problem

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

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


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

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

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

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


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

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

    How to continue from here?


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

    Tonio
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