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    Vecor Spaces and Linear Combinations

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    Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R

    Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.

    1. Label the following statements as true or false

    a) The zero vector is a linear combination of any nonempty set of vectors
    b) If S is a subset of a vector space V, then span (S) equals the intersection of all subspaces of V that contain S..

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    • Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R
    • Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.
    1. Label the following statements as true or false

    a) The zero vector is a linear combination of any nonempty set of vectors

    True: take the linear combination with all the coefficients =0

    b) If S is a subset of a vector space V, then span (S) equals the intersection of all subspaces of V that contain S..

    True. It is clear that if W is a subspace containing S, then it contains span(S) ...

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