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Proving linear transformation

  1. Feb 24, 2004 #1
    How will I prove that...
    Show that L: V -> W is a linear transformation if and only if
    L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any
    vectors u and v in V.

    For L(au +bv), this is my proof. (Is this wrong?)

    L(au + bv) = L [ a(a', b', c') + b(a'', b'', c'')]
    = L [ aa' + ba'', ab' + bb'', ac' + bc'' ]
    = (aa' + ab' + ac') + ( ba" + bb" +bc")
    = a(a' + b' +c') =b(a" + b" + c")
    = aL(u) + bL(v)
     
  2. jcsd
  3. Feb 24, 2004 #2
    I've never studied what you're doing formally, but I've always thought that what you're "proving" is the definition of a linear transformation. Why would you have to prove an arbitrary definition?

    cookiemonster
     
  4. Feb 24, 2004 #3
    Here's...

    Yeah that's right. It's an arbitrary definition of the linear transformation. My professor wants me to do it...

    In the textbook I'm using, it looks like this

    1. L(u+v) = L(u) + L(v)
    2. L(ku) = kL(u)
     
  5. Feb 24, 2004 #4
    Well, since it's an arbitrary definition, I don't really see the point of "proving" it.

    The only thing I can imagine him doing is asking you to combine the two conditions as it's usually stated. I usually see it in this form:

    [tex]L[\boldsymbol{v_1}+\boldsymbol{v_2}] = L[\boldsymbol{v_1}]+L[\boldsymbol{v_2}][/tex]
    [tex]L[a\boldsymbol{v}] = aL[\boldsymbol{v}][/tex]

    My guess is that he wants to see you combine these.

    Edit: Just noticed you typed the form yourself. Guess I should read a little more slowly next time...

    cookiemonster
     
  6. Feb 25, 2004 #5

    turin

    User Avatar
    Homework Helper

    I'm with cookiemonster. There is no such thing as proving a definition aside from showing the entry in a dictionary.
     
  7. Feb 25, 2004 #6

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    Well, as has been pointed out,

    "L: V -> W is a linear transformation" means
    1. L(u+v) = L(u) + L(v)
    2. L(ku) = kL(u)
    for all vectors u and v in V and scalars k.

    The problem is asking you to prove

    L: V -> W is a linear transformation

    if and only if

    L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any
    vectors u and v in V.


    So, you start with the assumption that "L: V -> W is a linear transformation" then prove "L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any vectors u and v in V."

    Then, (as a seperate piece of work!) you start with the assumption "L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any vectors u and v in V." and prove "L: V -> W is a linear transformation".
     
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