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Vector spaces, subspaces, subsets, intersections

  1. Mar 2, 2008 #1
    1. The problem statement, all variables and given/known data
    Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X[tex]\subseteq[/tex]Y. Show that Y[tex]\cap[/tex](X+Z) = X + (Y[tex]\cap[/tex]Z). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)


    2. Relevant equations



    3. The attempt at a solution

    Can anyone get me started on this one?
     
  2. jcsd
  3. Mar 3, 2008 #2

    CompuChip

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    Suppose that you have an element [itex]p \in Y \cap (X + Z)[/itex]. Then [itex] p \in Y[/itex] and [itex]p \in X + Z[/itex]. The latter means we can write [itex]p = x + z[/itex] with [itex]x \in X, z \in Z[/itex]. Now do you see how you can also write it as [itex]x' + y'[/itex] with [itex]x' \in X, y' \in Y \cap Z[/itex]?
     
  4. Mar 3, 2008 #3

    HallsofIvy

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    The hint looks like all you need. Have you tried that at all? Suppose v is in [itex]Y\cap(X+Z)[/itex]. That means it is in y and it can be written v= au+ bw where u is in X and w is in Z. Now you need to show that v is in [itex]X+ (Y\cap Z)[/itex]. That is, that it can be written in the form au+ bw where u is in X and w is in [itex]Y\cap Z[/itex]. Once you have done that turn it around: if v is in [itex]X+ (Y\cap Z)[/itex], can you show that it must be in [itex]Y\cap (X+ Z)[/itex]?
     
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