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In case of a linear dependent set of vectors, I know that there exist a vector which can be written as a linear combination of others. But, is it true that any vector in that set can be written as a linear combination of others. I think it is not.

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You are correct. Try $u=v$ and $w$ not parallel to them.

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You're correct. Intuitively, just because one of the vectors in a linearly dependent set $S = \{v_1,\ldots,v_n\}$ is in the subspace spanned by some of the other vectors in $S$ doesn't mean that EVERY vector in $S$ belongs to that subspace.

For example $t = (1,0,0)$, $u = (0,1,0)$, $v = (0,0,1)$, and $w = (0,1,1)$ form a linearly dependent set of vectors in $\mathbb{R}^3$ since $w = u+v$, but $t$ cannot be expressed as a linear combination of the other vectors. You can see this geometrically, since $t$ is not in the subspace spanned $u$, $v$, and $w$. (Of course you can also easily see this algebraically, but it's good to have a picture in mind!)

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I think you mean "since t is not in the subspace spanned by u, v and w." –  Jodles Aug 15 '13 at 3:33
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@Jodles: Whoops, that I did! :) –  Dan Aug 15 '13 at 4:21

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