Let v, w be elements of a vector space and assume v does not equal 0. If v, w are linear dependent, show that there is a number x such that w=xv. Thanks for any help, Mark.

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- Feb 15th 2009, 06:53 AMmark talaricolinearly dependent scalars
Let v, w be elements of a vector space and assume v does not equal 0. If v, w are linear dependent, show that there is a number x such that w=xv. Thanks for any help, Mark.

- Feb 15th 2009, 09:59 AMPlato
By definition of dependence $\displaystyle \alpha v + \beta w = 0\;\& \,\alpha \ne 0 \vee \beta \ne 0$.

Then if $\displaystyle \beta = 0$ then $\displaystyle \alpha v = 0$.

But $\displaystyle \alpha \ne 0\, \Rightarrow \,v = 0$ which is a contradiction.