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If Span(S)≠0 then S is linearly independant.

If Span(S)=0 then S is linearly dependant.

With S being a subset of V.

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- Thread starter sid9221
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If Span(S)≠0 then S is linearly independant.

If Span(S)=0 then S is linearly dependant.

With S being a subset of V.

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No. Span describes the set of all vectors in V that are linear combinations of vectors in S, it is entirely separate from linear independence. Linear independence means (there are various equivalent definitions) the following...

If the vectors**s**_{1}, s_{2}, ... s_{n} are linearly independent, then the equality **a**_{1}s_{1} + a_{2}s_{2} + ... + a_{n}s_{n} = 0 has only the trivial solution **a**_{1} = a_{2} = ... = a_{n} = 0 (where **a**_{1}, a_{2},... a_{n} are scalars). Equivalently, none of the vectors can be expressed as a linear combination of the others.

Take, for example, the vectors**[1 0 0]**, **[0 1 0]**, and **[0 0 1]** in R^{3}. These vectors are linearly independent, and yet they span all of R^{3} (the term for such a set of vectors is a *basis* for the space R^{3}).

If the vectors

Take, for example, the vectors

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Fredrik

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