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I'm having a very difficult time understanding a basis. I understand that it's basically just coordinates that span a subspace, but can someone help me to understand how a set can span a subspace and be a linearly independent set?

If a set spans a subspace doesn't this mean that any vector can be represented by a linear combination of the vectors in the subspace? If that is the case, isn't this a direct contradiction to the set being linearly independent?

What am I missing here?

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That the given vectors span the (sub)space, means that every evector in it can be written as at least one linear combination in the given vectors.

That the given vectors are linearly independant, means that no vector can be written in more than one way as linear combination.

Thus for a basis, every vector in the subspace can be written as exactly one linear combination of the basis vectors.

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    $\begingroup$ So a basis is the set of vectors that allow other vectors in the subspace to be represented by a linear combination of vectors in the basis? Am I totally off here? I'm sorry if what I'm writing is confusing, I don't know how to completely explain my confusion. $\endgroup$ – hax0r_n_code May 17 '14 at 14:22
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If a set spans a subspace doesn't this mean that any vector can be represented by a linear combination of the vectors in the subspace? If that is the case, isn't this a direct contradiction to the set being linearly independent?

I am not sure exactly what your complaint is here, but it sounds like you are saying, "since every vector is a linear combination of the basis vectors, doesn't that mean $0$ is a linear combination also, contradicting linear independence?" Well, sure, $0$ is a linear combination, but it is the following linear combination: $$ 0 b_1 + 0 b_2 + 0b_3 + \cdots + 0 b_n $$ where $b_i$ are the basis. So this does not contradict linear independence.

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