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    Vector Space and Subspace

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    Define vectors pace and subspace with examples.
    State and prove a necessary and sufficient condition for a subset of vectors to be a subspace.
    Show that the intersection and union of two sub spaces are also sub spaces.

    © BrainMass Inc. brainmass.com October 10, 2019, 8:25 am ad1c9bdddf
    https://brainmass.com/math/linear-algebra/vector-space-subspace-625063

    Solution Preview

    The solution illustrates the detailed explanation of the following:
    i) Definition of vectors pace and subspace with examples.
    ii) A necessary and sufficient condition for a subset of vectors to be a subspace.
    iii) Nature of the intersection and union of sub spaces.

    Vector Space
    A non-empty set of vectors V is called a vector space over a scalar field when the vector addition and scalar multiplication operations satisfy the following properties.
    A1) for all
    A2) for all
    A3) for every
    A4) There exist an element such that for all
    A5) For each , there is an element such that
    M1) for all and
    M2) for all and
    M3) for all and
    M4) for all and
    M5) ...

    Solution Summary

    The solution provides the definition and properties of vector space and subspace.

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