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Linearly Independence and Sets of Functions

  1. Feb 6, 2012 #1
    1. The problem statement, all variables and given/known data

    2012_02_06_5_29_39_PM.png

    3. The attempt at a solution
    I don't think I'm really understanding this problem. Let me tell you what I know: A set is linearly independent if [itex]a_1 A_1 +...+a_n A_n = \vec0[/itex] for [itex]a_1,...,a_n \in R[/itex] forces [itex]a_1 = ...=a_n = 0[/itex]. If [itex]f,g,h[/itex] take any of the [itex]x_i \in S[/itex], then one of the [itex]f,g,h[/itex] will map that element of S to zero, and no set containing [itex]\vec0[/itex] can be linearly independent.

    Somehow, however, I feel like I'm supposed to be arriving at the opposite conclusion.
     
  2. jcsd
  3. Feb 6, 2012 #2

    HallsofIvy

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    Okay, so these will be dependent if it is possible to have Af(x)+ Bg(x)+ Ch(x)= 0 for all x with at least one of A, B, C not equal to 0. Using the information given, [itex]Af(x_1)+ Bg(x_1)+ Ch(x_1)= A(0)+ B(1)+ C(1)= B+ C= 0.[/itex] Use the information given about [itex]x_2[/itex] and [itex]x_3[/itex] to get two more equations and solve for A, B, C.

    It is certainly true that no set containing the 0 vector is independent but that is irrelevant to this problem since none of f, g, or h is the 0 vector.
     
    Last edited by a moderator: Feb 6, 2012
  4. Feb 6, 2012 #3
    So, [itex]\vec0 = Af(x_2) + Bg(x_2) + Ch(x_2) = A +C[/itex] and [itex]\vec0 = Af(x_3)+ Bg(x_3) + Ch(x_3) = A+ B[/itex]. So we have equations [itex]A+B = B+C =A+C = 0[/itex], which necessitates that [itex]A=B=C=0[/itex]. Thus, the only way to get the set vector from a linear combination of f,g,h is with all-zero coefficients, meaning that the set is linearly independent.
     
  5. Feb 6, 2012 #4

    Bacle2

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    You may want to formalize things some more if you feel confused, by formally making the space of (Real, I assume)-valued functions, into a vector space, choosing a basis and then testing.
     
  6. Feb 6, 2012 #5
    Okay. I've made my discussion more "formalized" in my work. Thank you. Can I also get some help with this problem:

    Screen_shot_2012_02_06_at_7_14_08_PM.png

    Here's my work so far:

    I know that a linear transformation must have the property: Let T take elements of V to elements of W such that T(A) +T(B) + T(A+B) for all A,B in V. Thus, I said, let c,f be elements of S. I need (I'll use "ψ" as my special character indicating the transformation in the problem),

    ψ(T)(c) + ψ(T)(f) = T(χc) + T(χf) = T(χc + χf) = ψ(T)(c+f), so it passes addition. (Is that right? I'm not sure.) I also need rT(A) = T(rA) for all r in R. I write,

    ψ(T)r(c) = T(r(χc)) = rT(χc) = r ψ(T)(c).

    I'm not really sure that I'm doing these correctly. It would be most helpful if you could tell me what in particular is wrong with the approaches I attempt. That way, I think I will know both why the right way is right, and why the wrong way is wrong. Thanks!
     
  7. Feb 6, 2012 #6

    Bacle2

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    Just to make sure: is L(Fun(S),R) the vector space of linear maps from Fun(S) into R?

    If this is so, it may help re showing linearity.

    And if you can show that your map is a linear bijection between vector spaces of the same

    dimension, that shows you have an isomorphism.
     
    Last edited: Feb 6, 2012
  8. Feb 6, 2012 #7

    Bacle2

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    Actually, ψ is described as being linear in L(Fun(S),ℝ), not in S.

    But you cannot declare it to be linear; linearity follows from the properties
    of L(Fun(S),ℝ ), as well as from the definition of ψ .
     
    Last edited: Feb 7, 2012
  9. Feb 7, 2012 #8

    Deveno

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    this problem is more complicated then it looks. let me help you "un-wrap" it:

    φ:L(Fun(S),R)→Fun(S)

    so the input for φ is an element of L(Fun(S),R), such as T:

    φ:T→φ(T).

    now φ(T) is in Fun(S), which means it takes elements of S to elements of R:

    φ(T):s→φ(T)(s) = T(χs),

    where χs is the characteristic function of s

    (this would make more sense to me if it were χ{s}, as i am used to seeing characteristic functions defined on sets).

    to show φ is linear, you need to show that if T and T' are 2 elements of L(Fun(S),R):

    φ(T+T') = φ(T) + φ(T')
    φ(cT) = cφ(T).

    these are equalities of functions, so to prove they are equal functions, you need to show that for every s in S:

    φ(T+T')(s) = φ(T)(s) + φ(T')(s)
    φ(cT)(s) = cφ(T)(s).

    to evaluate the functions given above, you determine if:

    (T+T')(χs)

    and T(χs) + T'(χs) are equal,

    and similarly for the second equation.
     
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