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Introduction to Set Theory (precursor to better evaluation of LA)

  1. Jul 20, 2013 #1
    My goal: To show the dimension of space [itex]L[/itex] equals the length of any maximal flag of [itex]L[/itex];

    Is the following valid?

    My attempt:

    Let [itex]M \rightarrow {L_{i-1}, ... L_i}[/itex]

    where [itex]{e_i} \in L_i[/itex] [itex]|[/itex] [itex]e_i \not\in L_{i-1}[/itex]

    Assuming [itex]e_i \in M[/itex] and [itex]e_i \not\in L_{i-1}[/itex],

    we can say: [itex]e_i \in L_i[/itex] and [itex]L_i = M[/itex].

    Thus: [itex]{e_1, ... ,e_i} , {e_i} \in L_i = M[/itex] \ [itex]L_{i-1}[/itex]

    for [itex]n = dim L[/itex] or "finite dimension" of [itex]L[/itex] such that: [itex]L_o \subset L_1 \subset L_2 ... \subset L_n[/itex]
  2. jcsd
  3. Jul 20, 2013 #2
    What are ##M## and ##L_i##?
  4. Jul 20, 2013 #3
    I apologize for posting in a rush.

    [itex]L[/itex] is a maximal flag defined by [itex]L_0 \subset L_1 \subset L_2 ...[/itex] and [itex]L_i[/itex] is a space for which ([itex]{e_1, ... ,e_i}[/itex]) forms the basis. Assuming
    ([itex]{e_i, ... ,e_{i-1}}[/itex]) is valid, [itex]M[/itex] is a linear span of the aforementioned basis of [itex]L_i[/itex].
  5. Jul 21, 2013 #4
    Help... anyone? :(
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