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Math Help - Algebra - Vector Spaces

  1. #1
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    Algebra - Vector Spaces

    In the following, Mn,n(R) denotes the vector space of real
    n n matrices and, for A Ɛ Mn,n(R), A^T denotes the transpose of A.

    1. For each of the following, either use the subspace test to show that the given
    subset,
    W, is a subspace of V or explain why the given subset is not a subspace
    of
    V .

    (a)
    V = M2,2(R) and W = {A Ɛ V | A^T = A}.
    (b) V = R^3 and W = {(2t, 3t,5t) | t ƐR}.

    Any ideas on how to start? I'm useless with Vector spaces and am trying to learn it now, but i really don't know how to solve the above. Please be of guidance and assisstance.
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  2. #2
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    Dear James1220,

    you must prove if you take two element from W (mark: w1 and v2) than their linear comination w belong to W also. Linear combination means p*w1 + q*w2 where p and q arbitary real (or scalar). The operate * and + means same as in the original vector space.

    For exapmle. Let V = R^3 is a vector space. We can express an element as a triplet (x, y, z).
    Prove that elemnts with condition z=0 is subspace.
    Proof: Let w1 and w2 element of W. So we can write
    w1 = (x1, y1, 0) and w2 = (x2, y2, 0).

    Image the linear combination:
    w = p*w1 + q*w2 = p*(x1, y1, 0) + q*(x2, y2, 0) = (p*x1+q*x2,  p*y1+q*xy, 0)
    Is w element of W? Yes, of course since the 3th koordinate is null again.
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  3. #3
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    Thank you. So i'm guessing the fact that they are null, it means V is linearly independent? Is this the solution to part (b)?

    How would i do question (a), which has information on the transpose of A?
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  4. #4
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    Thank you. So i'm guessing the fact that they are null, it means W is a subspace of V??? Is this the solution to part (b)?

    How would i do question (a), which has information on the transpose of A?
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  5. #5
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    (A + B)^T = A^T + B^T (check out on a concrate matrix)

    You said: "they are null"
    What is null? You must take two element: w1 = (2t, 3t, -5t) and w2 = (2s, 3s, -5s) and go on...
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