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Linear algebra

  1. Feb 2, 2007 #1
    I need some help solving this...not even sure how to start...

    Let L:R(4) goes to R(4) be the linear transformation defined by
    -matlab notation, the value is a 4x1 column
    L ( [ a b c d])=[ a-b
    0
    c-d
    0 ]


    Show directly L is linear.
     
  2. jcsd
  3. Feb 2, 2007 #2

    radou

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    Homework Helper

    In order to get help, you should show some efforts.

    What is the definition of a linear operator?
     
  4. Feb 2, 2007 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    All I can say is "Just do it!"

    Write down the requirements for a linear transformation, insert your L, and do the calculations!
     
  5. Feb 2, 2007 #4
    Alright...

    I am still trying to figure out this message board...
    I forgot to include what I already know...

    I know you have to prove that L(u+v)=L(u)+ L(v) and L(c*u)=c*L(u), but I don't understand how to set it up.

    I tried to separate it into a1 and a2, but just get confused...

    Do I have to place it in the standard matrix representation then solve?
     
  6. Feb 2, 2007 #5
    still working with it...
    if I set u=a1, b1, c1, d1 and v=a1, b1, c1, d1
    set L(u+v)=(this is what I get)

    [ (a1+a2)-(b1+b2)
    0
    (c1+c2)-(d1+d2)
    0 ]

    which equals L(u)+L(v)

    for L(c*u)=c*L(u)

    [c(a-b) c*0 c(c-d) c*0]
    which converts to c[a-b 0 c-d 0] which breaks down to c*L(u)


    Is the close?
     
    Last edited: Feb 3, 2007
  7. Feb 3, 2007 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    That's exactly it.

    (except that you said " I set u=a1, b1, c1, d1 and v=a1, b1, c1, d1" when you mean "I set u=a1, b1, c1, d1 and v=a2, b2, c2, d2". Doesn't your class or textbook have some convention for writing vectors- say (a, b, c, d) or <a, b, c, d> rather than just a, b, c, d which can be confusing?
     
  8. Feb 3, 2007 #7
    ok...think i get it

    When I first did it, it didn't look right...almost too simple to be correct. Thanks for correcting me with my notation.

    my text does have a format but this doesn't support the large brackets required for a 4X1 matrix.

    I will try to make the matrix a little more easier to read...

    Would the notation for MATLAB entry suffice?
     
  9. Feb 3, 2007 #8

    radou

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    Homework Helper

    You can always write (a b c d)^T to represent a 4x1 matrix of you want.
     
  10. Feb 3, 2007 #9
    Check me

    Can someone verify my signs are correct for when I solved, with help from the board?
     
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