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If we say that $u$ and $v$ are linearly independent vectors in the vector space $V$ over some field $\Bbb F$, then $u+v$ is a linearly independent vector.

Why is this? Isn't it possible for $u+v=0_v$ where $0_v$ is the linearly dependent zero vector?

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But $u+v=0$ contradicts that $u,v$ are linearly independent. So $u+v\neq 0$ and you are done. –  1015 Feb 2 '13 at 16:02
    
A vector cannot be linearly independent. Linear independence is a property of sets of vectors. –  Chris Godsil Feb 2 '13 at 16:11
    
A vector can be linearly independent...just take the set containing only that vector. –  fretty Feb 2 '13 at 16:21
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First of all by definition, every non-zero vector is linearly independent. Now, if $u, v$ are linearly independent, then there is no way that $u + v=0$, because the coefficients of the linear combination $u+v$ are equal to $1$ for both $u$ and $v$ and so by definition of linear independence we will have $1 \cdot u + 1 \cdot v \neq 0$.

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