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Math Help - find basis

  1. #1
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    find basis

    Find a basis for each of the subspaces.

    1. The subspace R^5 given by the solution space of the homogeneous system system

    2x_1+2x_2-x_3+x_5=0
    -x_1-x_2+2x_3-3x_4+x_5=0
    x_1+x_2-2x_3-x_5=0
    x_3+x_4+x_5=0.


    2. The subspace of R^6 given by
    W={(u,v,w,x,y,z) in R^6: u+w=v-x, v=y+z}
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  2. #2
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    Hello,

    2x_1+2x_2-x_3+x_5=0 (1)

    -x_1-x_2+2x_3-3x_4+x_5=0 (2)

    x_1+x_2-2x_3-x_5=0 (3)

    x_3+x_4+x_5=0. (4)

    (2)+(3) :

    -3x_4=0 \rightarrow x_4=0

    So the system is resumed to :

    2x_1+2x_2-x_3+x_5=0 (1)

    x_1+x_2-2x_3-x_5=0 (3)

    x_3+x_5=0. (4')

    (4') \rightarrow x_3=-x_5

    2x_1+2x_2+2x_5=0 (1')

    x_1+x_2+x_5=0

    Now, keep the value for x_4=0.

    Then replace x_5 by 0 for example and for 1. Then you will have x_3 thanks to (4')
    And for x_1+x_2 do the same !
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