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    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

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    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?
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    (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
    v \in V is a linear combination of elements of U if

    v = \alpha_1 v_1 + ... + \alpha_n v_n for some \alpha_1, ... , \alpha_n \in \mathbb{K} and v_1 , ... , v_n \in V

    \text{Sp}(U) is the set of all linear combinations of elements of U .

    \{ v_1 , ... , v_n \} is a spanning set for U if v = \alpha_1 v_1 + ... + \alpha_n v_n for all v \in V

    \text{Sp}(U) = V means U is a basis for V (if U is also linearly independent).
 
 
 
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