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Proof involving vector spaces and linear transformations

  1. Jan 28, 2009 #1
    1. Suppose V,W are vector spaces over a field F and that T: V ---> W is a linear transformation. Show that for any v belonging to V that T(-v) = -T(v)

    2. -T(v) denotes the additive inverse of T(v)

    3. I think i'm really overcomplicating it =/ But i have

    0v = T( v - v ) = T(v) + T(-v)

    Then add -T(v)

    0v + -T(v) = T(v) + T(-v)

    (T(v) + T(-v)) -T(v) = T(v) + T(-v)


    T(-v) = T(v) + T(-v)

    then i suppose it could go to

    T(-v) = 0v

    but that doesn't help i'm going round in circles. Basically i need a starting point.
  2. jcsd
  3. Jan 28, 2009 #2


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    Looks more like you need someone to tell you when to stop. :smile:

    When you obtain 0=T(v)+T(-v), you're done. If T(-v) has that property, it is the additive inverse of T(v). You should however explain more clearly why T(v+(-v))=0.

    Hm, are you allowed to use that 0x=0 for all x without proof? If not, it's easy to prove using the trick 0x=(0+0)x.
    Last edited: Jan 28, 2009
  4. Jan 28, 2009 #3

    Right so when i get:

    0v = T(v) + T(-v)

    i just move the T(v) over because 0v=0?


    -T(v) = T(-v)

    seems a bit too easy! I've already proved 0x=0 .

    Can you explain more about

    'You should however explain more clearly why T(v+(-v))=0.'

    please =]
  5. Jan 28, 2009 #4
    You shouldn't need to move anything over, you know that 0v = 0, therefore at the step where you have T(v) + T(-v) = 0 that means T(-v) is the additive inverse of T(v), remember what definition of additive inverse is!
  6. Jan 28, 2009 #5


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    I'm not sure if I can tell you anything new without giving you the complete solution. What you want to prove is that T(v)+T(-v)=0. And the proof of that goes like this:


    Can you fill in the missing steps?
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