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Math Help - Basis of a Linear Manifold

  1. #1
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    Basis of a Linear Manifold

    Given the definition of a linear manifold as a set M of undefined elements "X" and scalars "a" from a real field F; and an association X+Y between elemets of M, and an association aX between elements F and M, such that:

    X+Y = Y+X
    a(X+Y) = aX + aY
    etc

    How do you prove the existence of elements of M for which aX + bY + cZ +.. = 0 has no solution other than a = b = c = ... = 0. Or, more basically, elements X and Y such that aX +bY= 0 has no solution other than a = 0 and b= 0.
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  2. #2
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    A Basis

    Consider aX + bY = 0
    Either a Y exists st aX + bY = 0 only if b=0, in which case X and Y are Linearly Independent, or Y = pX for all Y and M is at least one-dimensional.

    If X and Y are LI, consider aX + bY + cZ = 0.
    Either a Z exists st aX + bY +cZ = 0 only if c=0, in which case X, Y and Z are LI, or Z = pX +qY for all Z and M is at least two-dimensional.

    Eventually you reach a point where no W exists st aX + bY + cZ +....+ dU + eW = 0 only if e = 0, and then every W can be expressed as a linear combination of X, Y, Z,....,U which is a basis of M of dimension r (number of LI members X,Y,Z....,U).

    "Vector" was never mentioned.
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    I would like to reword original question.

    Can you prove from the definition of a linear manifold that linearly independent vectors must exist?

    The standard definition of linear independence is: if aX+bY+cZ+..=0 only has a solution for a=b=c..=0, then X,Y,Z.. are linearly independent. Is that an existence proof?
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  4. #4
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    Quote Originally Posted by Hartlw View Post
    The standard definition of linear independence is: if aX+bY+cZ+..=0 only has a solution for a=b=c..=0, then X,Y,Z.. are linearly independent. Is that an existence proof?
    No, that's just a definition. For a proof notice that aX=0 then a=0 or X=0 (by the properties of being a linear manifold) so any nonzero X we have \{ X\} is l.i. now take S= \{ A\subset M : A\  \mbox{is linearly independent} \} and use Zorn's lemma to guarantee the existence of a maximal l.i. set.

    The argument you use in your second post works if your manifold is finite dimensional (a fact one does not know before actually finding a basis).
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