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I am new to advanced linear algebra so please excuse the problem goes:

Let S be a subset of a vector space V , such that S is also contained in a subspace W of V . If S is linearly independent as a subset of W, can it be dependent as a subset of V ?

My thoughts are that it does not matter if you are in a subspace or the vector space itself, the set will be linearly dependent or independent regardless. I can't think of any example that would say otherwise... Hints are greatly appreciated.

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    $\begingroup$ Linear dependence of a subset depends only on it's relationship with the field over which the vector space is, not on what subspace it is sitting in, since the linear combinations are taken with scalars from the field and elements from the subset. $\endgroup$ – астон вілла олоф мэллбэрг Sep 15 '17 at 4:52
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You are right. The definition of linear independence of a set $S$ depends only on whether linear combinations of members of $S$ are or are not $0$. It is the same for any vector space that contains $S$ (with, of course, the same operations of addition and scalar multiplication on the members of $S$ and their linear span).

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The operation of vector addition and scalar multiplication result in the same answer irrespective of whether they are regarded as elements of $V$ or the subspace $W$. (That is essentially the condition for a subset being a vector subspace!)

To give an analogy unrelated to mathematics, imagine a tennis situation: The commentator might say game, set, and match for a player when a point is won. However, what the player has to do remains the same whether it is to win a point, a game, a match or the championship: just keep sending the ball across to the opponent's court.

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