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    Matrices, Vector Spaces and Subspaces

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    Give a demonstration as to why or why not the given objects are vector subspaces of M22

    a) all 2 X 2 matrices with integer entries

    A vector space is a set that is closed under finite vector addition and scalar multiplication.

    It is not a vector space, since V is NOT closed under finite scalar multiplication. For instance, take a 2 by 2 matrix with integer entries

    Then is NOT in V since not every entry is an integer.

    b) all matrices a b

    c d such that: a + b + c + d = 0

    It is a vector space, since V is closed under finite vector addition and scalar multiplication.

    (1) V is closed under finite vector addition

    For instance, take any two 2 by 2 matrices in V

    and
    where and . Obviously, we have
    =
    as

    And
    (2) V is closed under finite scalar multiplication.

    as

    c) all 2 X 2 matrices A such that det(A) = 0

    It is NOT a vector space, since V is NOT closed under finite vector addition. For instance, take TWO 2 by 2 matrices with det(A)=det(B)=0

    and
    BUT
    with det(A+B)=2*2-1*1=3
    So,

    Please see the attached file for the fully formatted problems.

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    https://brainmass.com/math/matrices/matrices-vector-spaces-and-subspaces-105556

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