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Math Help - Subspace of R3

  1. #1
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    Subspace of R3

    I know that for a set u of vectors to be called a subspace in R^n, it must satisify the conditions:

    1- 0 E u
    2- x, y E u --> x+y E u
    3- x E u --> ax E u (a E R)

    But I still cant manage to determine which sets are a subspace for R^n..

    an example of a question is:

    Which of the following sets are subspaces of R^3?

    A. {(-3x + 5y, 8x +4y, -2x-4y)| x, y arbitrary numbers}
    B. {(x,y,x)| x<y<z}
    C. {(6,y,z)| y, z arbitrary numbers}
    D. {(x,0,0)| x arbitrary number}
    E. {(x,y,z)| x+y+z = 3}
    F. {(x,y,x)| 6x+9y-7z=0}

    The answer I found was:
    a
    c
    d
    f


    but apparently it's wrong...so I don't know


    Another question is:

    Which of the following subsets of R^(3x3) are subspaces of R^(3x3) ?

    A. The invertible matrices
    B. The matrices whose entries are all integers
    C. The symmetric matrices
    D. The matrices whose entries are all greater than or equal to
    E. The matrices with all zeros in the third row
    F. The diagonal matrices

    Wouldn't it be all because they're all subsets of R^(3x3) ??


    Could someone kindly explain to me how to find the answers, and what the answer would be to that question?




    This question has to do with linear dependency:
    Which of the following sets of vectors are linearly independent?

    A. { ( -2, 8, 0 ), ( 7, -9, 0 ), ( 6, -5, 0 ) }
    B. { ( 3, -4, -7 ), ( 2, -3, -8 ), ( 1, -1, 1 ) }
    C. { ( -9, 6 ) }
    D. { ( -7, 2 ), ( -3, -8 ) , ( 4, 5 ) }
    E. { ( -6, -1, 5 ), ( -1, 9, 4) }
    F. { ( 3, -4, -7 ), ( 2, -3, -8 ), ( 1, 1, -1 ), ( 3, 11, -1 ) }
    G. { ( -8, -3, -1, 5 ), ( -1, 5, -3, 2) }
    H. { ( -3, 0, 2 ), ( 6, 5, -4 ), ( -1, -8, 9 ) }

    the answer I get are B, G and H..but it still shows up as wrong..

    Thanks in advance!

    HopefulMii
    Last edited by HopefulMii; December 14th 2008 at 07:58 AM.
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  2. #2
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    Quote Originally Posted by HopefulMii View Post
    A = {(-3x + 5y, 8x +4y, -2x-4y)| x, y arbitrary numbers}
    I do this first one.

    Let \bold{x} = (-3x_1 + 5x_2, 8x_1 + 4x_2, -2x_1 - 4x_2) and \bold{y} = (-3y_1 + 5y_2, 8y_1 + 4y_2, - 2y_1 - 4y_2).

    This means, \bold{x} + \bold{y} = ( - 3(x_1 + y_1) + 5(x_2+y_2), 8(x_1+y_1) + 4(x_2+y_2), - 2(x_1+y_1) - 4(x_2+y_2) ) \in A

    And, k\bold{x} = (-3(kx_1) + 5(kx_2), 8(kx_1) + 4(kx_2), - 2(kx_1) - 4(kx_2))\in A.

    Therefore A is a vector space.
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  3. #3
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    okay..

    but how do I prove that for the ones that are just like:

    {(x,y,z)}| 6x + 9y -7z=0} ?

    What I think I'm supposed to do is isolate z

    so z = (6x+9y)/7

    and then I have the vector (x,y, (6x+9y)/7)

    but then from there I don't know what to do...
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  4. #4
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    Quote Originally Posted by HopefulMii View Post
    okay..

    but how do I prove that for the ones that are just like:

    {(x,y,z)}| 6x + 9y -7z=0} ?
    If 6x_1 + 9x_2 - 7x_3 = 0 and 6y_1 + 9y_2 - 7y_3 = 0.
    This means (their sum) 6(x_1+y_2) + 9 (x_2 + y_2) - 7 (x_3 + y_3) = 0.
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