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The direct sum of subspaces

  1. Jan 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi, everybody!
    I'd like to ask you about the direct sum of subspaces...
    I refer to two linear algebra books; 1)Friedberg's book, 2)Hoffman's book.
    First of all, I write two definitions of direct sum of subspaces...
    in the book 1),
    Def.1). Let W[itex]_{1}[/itex],...,W[itex]_{k}[/itex] be subspaces of a vector space V. We call V the direct sum of the subspaces W[itex]_{1}[/itex],...,W[itex]_{k}[/itex]
    and write V=W[itex]_{1}[/itex][itex]\oplus[/itex]...[itex]\oplus[/itex]W[itex]_{k}[/itex], if
    V = [itex]\sum[/itex][itex]^{k}_{i=1}[/itex]W[itex]_{i}[/itex] and
    W[itex]_{j}[/itex][itex]\cap[/itex][itex]\sum[/itex][itex]_{i\neq j}[/itex]W[itex]_{i}[/itex]={0} for each j(1[itex]\leq[/itex]j[itex]\leq[/itex]k);I intepret this condition by logical form as follows;[itex]\forall[/itex]j(1[itex]\leq[/itex]j[itex]\leq[/itex]k and j is an integer [itex]\rightarrow[/itex]W[itex]_{j}[/itex][itex]\cap[/itex][itex]\sum[/itex][itex]_{i\neqj}[/itex]W[itex]_{i}[/itex]={0}).

    In the book 2),
    Def.2). Let W[itex]_{1}[/itex],...,W[itex]_{k}[/itex] be subspaces of a finite-dimensional vector space V. We call V the direct sum of the subspaces W[itex]_{1}[/itex],...,W[itex]_{k}[/itex]
    and write V=W[itex]_{1}[/itex][itex]\oplus[/itex]...[itex]\oplus[/itex]W[itex]_{k}[/itex],
    if V = [itex]\sum[/itex][itex]^{k}_{i=1}[/itex]W[itex]_{i}[/itex] and the subspaces have
    the property such that for each j, 2[itex]\leq[/itex]j[itex]\leq[/itex]k, we have W[itex]_{j}[/itex][itex]\cap[/itex](W[itex]_{1}[/itex]+...+W[itex]_{j-1}[/itex])={0};I interpret this condition by logical form as follows; [itex]\forall[/itex]j(2[itex]\leq[/itex]j[itex]\leq[/itex]k and j is an integer [itex]\rightarrow[/itex]W[itex]_{j}[/itex][itex]\cap[/itex](W[itex]_{1}[/itex]+...+W[itex]_{j-1}[/itex])={0}).

    Now, as you can see, in Def.1), since [itex]\sum[/itex][itex]_{i\neqj}[/itex]W[itex]_{i}[/itex] does not exist when k=1, I cannot determine whether direct sum of subspaces is defined or not when k=1.
    On the other hand, in the second definition; that is, Def.2), from the logical form, I can see that it is vacuously true when k=1. So, in this case, I can say V is a direct sum of its subspace V=W[itex]_{1}[/itex]. I think this means that direct sum of subspaces can be defined even when k=1.
    How do you think about Def.1)? Do you think that there're some mistakes in the reasoning I've suggested above?
    To sum up, I wanna ask you if 1. direct sum can be defined even when k=1
    2. if so, where do I make such mistakes??

    I hope you help me solve this problem...
    Thank you for reading my long questions...
    Have a nice day!!!

    (If you want to have more specific information of definitions of them, refer to p. 275 in 1) and p. 219~220 in 2).)
    2. Relevant equations
    Red lines above indicate
    i [itex]\neq[/itex] j
    3. The attempt at a solution
     
    Last edited: Jan 16, 2012
  2. jcsd
  3. Jan 16, 2012 #2

    jbunniii

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    You're correct that the first definition doesn't make sense for k = 1 as written. It would make sense if you make the further (reasonable) assumption that

    [tex]\sum_{i \neq j} W_i = \{0\}[/tex]

    when there are no [itex]i \neq j[/itex]. Under this assumption, the second condition is vacuously true when k = 1.

    To answer your questions,

    1. Yes, the direct sum is defined even when k = 1, but in that case it simply reduces to [itex]V = W_1[/itex], which isn't very interesting. If k= 1, you don't need the second part of either definition because there is only one subspace under consideration.

    2. You didn't make a mistake. You noticed a technical error in the first definition, which means you were reading carefully. It's good to check whether definitions make sense in all cases. Unfortunately, it's not uncommon for authors to be a bit sloppy in ways similar to this.
     
    Last edited: Jan 16, 2012
  4. Jan 17, 2012 #3
    I really appretiate you for replying question.
    That's what I want to hear from you as the answer for my questions.
    Thanks again!!!
     
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