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Homework Help: Linear Algebra - Find a subspace

  1. Mar 2, 2012 #1
    1. The problem statement, all variables and given/known data

    V is a subspace of the vector space [itex]R^{3}[/itex] given by:
    V = {(x, y, z) E [itex]R^{3}[/itex] / x + 2y + z = 0 and -x + 3y + 2z = 0}
    Find a subspace W of [itex]R^{3}[/itex] such that [itex]R^{3}[/itex] = V[itex]\oplus[/itex]W

    I'm really lost in this. My teacher didn't give any example of how to solve this kind of exercise... Can anyone help me how to start and develope this?
  2. jcsd
  3. Mar 2, 2012 #2


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    each of those equations defines a plane - what is the intersection of two planes?
  4. Mar 2, 2012 #3
    A straight line?

    Well, if I solve this system:

    x = -2y + z
    y = -3/5z --> x = -11/5z
    Last edited: Mar 2, 2012
  5. Mar 2, 2012 #4
    To start with, do you know what your looking for ? Do you know what is the direct sum of two subspaces?
  6. Mar 2, 2012 #5


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    So the subspace V is a line through the origin, and can be represented by the span of a single vector parallel to that line.

    Now onto Dansure's question...
  7. Mar 2, 2012 #6


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    W will be a plane perpendicular to that line, containing the origin.
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