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Linear Algebra- Transformations and

  1. Oct 13, 2009 #1
    1.
    Question:
    Which of the following linear transformations T from |R^3 to |R^3 are invertible? Find The inverse if it exists.

    a. Reflection about a plane
    b. Orthogonal projection onto a plane
    c. Scaling by a factor of 5
    d. Rotation about an axis

    2. Relevant equations



    3. The attempt at a solution
    Well I know only a, c, and d are invertible but I do not know how to go about finding their inverses. Only i have assumed so far it involves some vector <x,y,z> about some arbitrary thing such as in part a, a plane through the origin with an arbitrary normal <a,b,c> but Im not even sure about that.
     
    Last edited: Oct 13, 2009
  2. jcsd
  3. Oct 13, 2009 #2

    Dick

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    Hi KyleS4562, welcome the Forums. I think they want a fairly abstract answer to each question, since they didn't describe most of them in enough detail to let you give a formula like <x,y,z> transforms to whatever. Just answer in the same spirit they asked. What's the inverse of a reflection? What's inverse of an expansion by 5? What's the inverse of a rotation? Just answer in words.
     
    Last edited: Oct 13, 2009
  4. Oct 13, 2009 #3
    Thanks... i have about a page and a have worth of random row reducing trying to come up with a generic inverse matrix for a with any plane with any given <a,b,c> normal and I still not got the final matrix so I like your answer better.
     
  5. Oct 13, 2009 #4

    Dick

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    Absolutely, now what is your answer? I'd like to see it to make sure we're on the same wavelength.
     
  6. Oct 13, 2009 #5
    A. Invertible. Lets call this transformation T which as a transformation matrix A. So since it is invertible there exists an A^-1 such that: T(<x,y,z>)= <x1,y1,z1> where A<x,y,z> = <x1,y1,z1> so that A^-1 <x1,y1,z1> = <x,y,z> for any given plane

    and something like that for the other ones or am I being to vague?
     
  7. Oct 13, 2009 #6

    Dick

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    No, you are being too specific. Isn't the inverse of a reflection the same reflection? Just say that, I know you know it.
     
    Last edited: Oct 13, 2009
  8. Oct 13, 2009 #7
    Wow I didn't think of that. So it is just A^-1 = A

    then for b we can actually do it since

    A= [5,0,0] [0,5,0] [0,0,5]

    A^-1 = [1/5,0,0] [0,1/5,0] [0,0,1/5]

    then d if a rotation around an axis is defined by theta degrees than the inverse matrix will rotate the vector 2pi-theta in the same direction
     
  9. Oct 14, 2009 #8

    Dick

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    Right, exactly. The inverse of scaling by 5 is scaling by 1/5. And the inverse of a rotation around an axis by theta is rotation around the same axis by -theta. Or as you put it 2pi-theta. But they are the same thing.
     
  10. Oct 14, 2009 #9
    Thank you very much for all your help
     
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