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set of vectors {v1, v2.......,vn} belongs to R^m where n

MIT 18.06 Linear Algebra, Spring 2010
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not necesarily. for instance \[ \large \{(1,2,3),(0,0,0)\} \] is a linearly dependent set of two vectors in \(\mathbb{R}^3\) and \(2<3\).
how to prove this?
since you can prove that, for at least one case this is not true, therefore it can not be generalized.

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To prove that (1,2,3) and (0,0,0) are dependent: Let us assume the contrary. If they are independent, then k1* (1,2,3) + k2 * (0,0,0) = (0,0,0) implies k1 = k2 = 0. But, in this case, only k1 has to be 0. k2 can be any number and the expression still holds. Hence the proof.

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