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Multilinear Algebra Definition

  1. May 1, 2013 #1
    I'm reading Lee's Introduction to Smooth Manifolds and I have a question on the definition of a multilinear function

    Suppose [itex] V_{1},...,V_{k}[/itex] and [itex]W[/itex] are vector spaces. A map [itex] F:V_{1} \times ... \times V_{k} \rightarrow W [/itex] is said to be multilinear if it is linear as a function of each variable separately when the others are held fixed: for each i,

    [itex] F(v_{1},...,av_{i} + a'v^{'}_{i},...,v_{k}) = aF(v_{1},...,v_{i},...,v_{k}) + a'F(v_{1},..., v^{'}_{i},...,v_{k}) [/itex]

    I'm thinking that it should look like this,

    [itex] F(u_{1},...au_{k} + a'v_{1},...,v_{k}) = aF(u_{1},...,u_{k})+a'F(v_{1},...,v_{k}) [/itex]
    any comments?
     
    Last edited: May 1, 2013
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  3. May 1, 2013 #2

    Office_Shredder

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    F on the left hand side takes in 2k-1 vectors, and on the right hand side just k.

    The definition in the book is correct, do you think it's a typo or are you just throwing out an idea for a different definition?
     
  4. May 1, 2013 #3
    different idea of course sorry, definitely not trying to correct him, i'm not seeing it as clearly, umm "F on the left hand side." Which F are you talking about? Lee's definition or what I'm thinking it should look like?
     
  5. May 1, 2013 #4

    Office_Shredder

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    OK just wanted to make sure. The F on the left hand side I'm talking about here.

    Let's suppose k=2. It looks like what you have written is

    [tex] F(u_1, a u_2 + a' v_1, v_2) = a F(u_1, u_2) + a' F(v_1, v_2) [/tex]

    And F on the left hand side of the equation, and on the right hand side of the equation, have a different number of variables as input
     
  6. May 1, 2013 #5
    Hmm, honestly I'm still a little confused, I'm trying to think of a more concrete example such as F being an inner product

    if we let V=ℝn for example, then V is a vector space over the field ℝ with the usual addition of vectors and scalar multiplication.

    so [itex] <\cdot,\cdot> : V \times V \rightarrow ℝ [/itex] given by [itex] \sum_{i=1}^{n} a_{i}b_{i} [/itex] is a multilinear function (namely a bilinear function)

    so if we let x,y,z, be in ℝn and let F = <.,.>, then F(ax+by,z) = aF(x,z) + bF(y,z)
    = a<x,z> + b<y,z> right?

    so using Lee's notation I'm not really seeing it.
     
  7. May 1, 2013 #6
    or is it better to see it this way?

    [itex]F(a(u_{1},...,u_{k}) + b(v_{1},...,v_{k})) = aF(u_{1},...,u_{k}) + bF(v_{1},...,v_{k})[/itex] ?
     
  8. May 2, 2013 #7

    micromass

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    No, that would imply that ##F## is linear.
     
  9. May 2, 2013 #8
    In the case k = 2, the definition says that
    [tex]F(a v_1 + a' v_1', v_2) = a F(v_1, v_2) + a' F(v_1', v_2)[/tex]
    and
    [tex]F(v_1, a v_2 + a' v_2') = a F(v_1, v_2) + a' F(v_1, v_2').[/tex]

    In other words, the map [itex]V_1 \to W,\;v_1 \mapsto F(v_1, v_2)[/itex] is linear (for fixed [itex]v_2[/itex]), and the map [itex]V_2 \to W,\;v_2 \mapsto F(v_1, v_2)[/itex] is linear (for fixed [itex]v_1[/itex]). This is what it means to be linear as a function of each variable separately.
     
  10. May 2, 2013 #9
    got it! thanks!
     
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