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# Definition of Span Watch

1. If you were to say U, a subset of V, spans V does this mean:
i) V is contained in (i.e. is a proper subset of) span(U)
or
ii) span(U)=V

Thanks
2. The span of any set of vectors (of a vector space) is a subspace. So is contained in the original vector space. Do you see how this makes sure that (i) cannot be a possibility?

To stress; if V is (properly) contained in Span(U) then there is some u in U, say u* = a.u + b.v + ... + c.w where a,b,...,c are scalars and u,v,...,w are elements of U (which is a subset of the vector space) that can not be an element of the vector space V. A vector space is closed under scalar multiplication, addition - so where would u* also have to be?
3. (Original post by Palabras)
If you were to say U, a subset of V, spans V does this mean:
i) V is contained in (i.e. is a proper subset of) span(U)
or
ii) span(U)=V

Thanks
is a linear combination of elements of if

for some and

is the set of all linear combinations of elements of .

is a spanning set for if for all

means is a basis for (if U is also linearly independent).

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Updated: December 25, 2010
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