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Proof Help Spans

  1. Sep 16, 2004 #1
    Proof Help!! Spans!!

    I was give the fallowing to prove and i would like it know if anyone thinks i did this right lol....

    Let [tex] u_1, u_2, ..., u_k [/tex] be vectors in [tex] R^n [/tex] and [tex] A [/tex] be a [tex] m x n [/tex] matrix. Prove that if [tex] v [/tex] is int he span of [tex] u_1, u_2, ..., u_k [/tex] then [tex] Av [/tex] is in the span of [tex] Au_1, Au_2, ..., Au_k [/tex].

    this is what i did to prove this...

    If [tex] v [/tex] is in the span of the set [tex] S [/tex] which follows [tex] S = {u_1, u_2, ...,u_k}[/tex] then there is set of coeffients that make the following linear combonation of the elements of the set [tex]S[/tex] that follows the linear equation to prove that [tex]v [/tex] is in the span of [tex] S[/tex]...
    [tex]c_1 u_1 + c_2 u_2 + ... + c_k u_k = v[/tex]

    to show that [tex]Av[/tex] is in the span of [tex]Au_1, Au_2, ..., Au_k [/tex]
    thus the fallowing linear equation must be true...[tex]c_1 Au_1 + c_2 Au_2 + ... + c_k Au_k = Av[/tex] factoring matrix [tex] A[/tex] gives [tex] A(c_1 u_1 + c_2 u_2 + ... + c_k u_k) \rightarrow Av[/tex] thus proving Av is in the span of [tex]Au_1, Au_2, ..., Au_k [/tex]

    so did i make any mistakes is this crap? let me know please....

    FOR ANYONE WHO'S LATEX ISN'T LOADING HERES THE RAW TYPING
    I was give the fallowing to prove and i would like it know if anyone thinks i did this right lol....

    Let u_1, u_2, ..., u_k be vectors in R^n and A be a m x n matrix. Prove that if v is int he span of u_1, u_2, ..., u_k then Av is in the span of Au_1, Au_2, ..., Au_k .

    this is what i did to prove this...

    If v is in the span of the set S which follows S = {u_1, u_2, ...,u_k} then there is set of coeffients that make the following linear combonation of the elements of the set S that follows the linear equation to prove that v is in the span of S...
    c_1 u_1 + c_2 u_2 + ... + c_k u_k = v

    to show that Av is in the span of Au_1, Au_2, ..., Au_k
    thus the fallowing linear equation must be true...c_1 Au_1 + c_2 Au_2 + ... + c_k Au_k = Av factoring matrix A gives A(c_1 u_1 + c_2 u_2 + ... + c_k u_k) -> Av thus proving Av is in the span of Au_1, Au_2, ..., Au_k

    so did i make any mistakes is this crap? let me know please....
     
    Last edited: Sep 16, 2004
  2. jcsd
  3. Sep 16, 2004 #2

    mathwonk

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    just tell me briefly what you are trying to do. i cannot digest all that crap.
     
  4. Sep 17, 2004 #3

    matt grime

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    the short answer is, yes, you've got it. (A is linear so it commutes with scalar multiplication and addition of vectors, that's all they're getting at in the question).
     
  5. Sep 17, 2004 #4

    mathwonk

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    if v = some linear combination of the u's, then since matrix multiplication commutes with linear combinations, then Av is the same linear combination of the Au's. QED.
     
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