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How to prove that the dim of two subspaces added together equals dim

  1. Mar 30, 2010 #1
    How to prove that the dim of two subspaces added together equals dim of their union plus 1 iff one space is a subest of the other

    In other words,
    subspaces: V, S of Vector space: W

    [tex] dim(V+S) = dim(V \cap S) +1 [/tex]

    if [tex]V \subseteq S[/tex] or [tex]S \subseteq V [/tex]
     
  2. jcsd
  3. Mar 30, 2010 #2
    Re: Prove

    I dont think this is true. If V=S then arent V+S and V intersect S the same?
     
  4. Mar 30, 2010 #3

    jbunniii

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    Re: Prove

    eok20 is correct, the statement is false.

    What is true is

    [tex]dim(V+S) = dim(V) + dim(S) - dim(V \cap S)[/tex]

    which reduces to a simpler form if [itex]V \subseteq S[/itex] or [itex]S \subseteq V[/itex].
     
  5. Mar 30, 2010 #4
    Re: Prove

    perhaps I mistyped the question.
    [tex] dim(V+S) = dim(V \cap S) +1 [/tex] is given for these subspaces.

    One has to prove that

    [tex] V \subseteq S [/tex] or [tex] S \subseteq V [/tex]
     
  6. Mar 30, 2010 #5

    jbunniii

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    Re: Prove

    OK, suppose

    [tex]V \not\subseteq S[/tex] and [tex]S \not\subseteq V[/tex].

    Then V contains at least one element not in S, hence at least one element not in [itex]V \cap S[/itex]. Thus [itex]dim(V) > dim(V \cap S)[/itex], and since dimensions are integers, this is the same as [itex]dim(V) \geq dim(V \cap S) + 1[/itex].

    Similarly, S contains at least one element not in V, so [itex]dim(S) \geq dim(V \cap S) + 1[/itex].

    Now apply those inequalities to

    [tex]dim(V+S) = dim(V) + dim(S) - dim(V \cap S)[/tex]

    to achieve a contradiction.
     
  7. Mar 30, 2010 #6
    Re: Prove

    How do I come to the fact that

    [tex] dim(V+S) = dim(V) + dim(S) - dim( V \cap S) [/tex]


     
  8. Mar 30, 2010 #7

    jbunniii

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    Re: Prove

    This is a standard result that should be in just about any linear algebra textbook. E.g., theorem 2.18 in Sheldon Axler's "Linear Algebra Done Right." See page 33 in this online preview:

    http://books.google.com/books?id=BN...esnum=1&ved=0CDoQ6AEwAA#v=onepage&q=&f=false"

    Here is another proof (PDF file), but it looks more longwinded than it needs to be:

    http://www.its.caltech.edu/~clyons/ma1b/intsumdimthm.pdf [Broken]

    [Edit]: See also Problem 16, parts 2 and 3 here:

    http://en.wikibooks.org/wiki/Linear_Algebra/Combining_Subspaces
     
    Last edited by a moderator: May 4, 2017
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