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Finding the kernel and range of a tranformation

  1. Dec 1, 2005 #1
    If L(x) = (x1, x2, 0)^t and L(x) = (x1, x1, x1)^t

    What is the kernel and range?
     
  2. jcsd
  3. Dec 1, 2005 #2

    HallsofIvy

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    You have defined two different functions. Which one are you referring to?

    Do you know the DEFINITION of "kernel"
     
  4. Dec 1, 2005 #3
    I need to do both. I know the kernel is basically setting to zero, but my book is awful on explaining stuff. I also know the range is basically solving for y and row reducing, but I am foggy on the presentation.
     
  5. Dec 1, 2005 #4

    JasonRox

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    I'll explain it to you, and I want to see you attempt to answer it.

    The range is simple. It's just like Calculus, sort of.

    For example, the function f(x)=x^2 has a domain of R(-infinite,infinite), and range (0,infinite).

    The range is simply all the possible vectors you can obtain from the transformation. If T(x) = (0,0), then all the possible vectors that come "out" of the transformation are in the set {(0,0)}, which is the range.

    What's the range of T(x)=(x,y)?

    The answer is R^2, which is any vector in the Cartesian Plane.

    The kernel of a transformation is the set of vectors that transform into the zero vector. So, for the first one T(x)=(0,0) is all the vectors, since all the them become a zero vector, so the answer is R^2.

    What's the kernel of T(x)=(x,y)?

    Well, the only possible vector that can transform into a zero vector is the zero vector itself.

    Note: I have no idea what you mean by exponent t. Maybe that signifies that it is a transformation... I have no idea.
     
  6. Dec 2, 2005 #5

    HallsofIvy

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    Your book is awful on explaining "setting to zero"?? Not a whole lot to explain is there?

    For your first function L(x1,x2,x3))= (x1, x2,0).
    Set that equal to 0: (x1, x2,0)= (0, 0, 0).

    What does that tell you about x1 and x2? What does it tell you about x3?

    Now the range: what do all possible values of L, that is all vectors of the form (x1, x2,0), have in common?
     
  7. Dec 2, 2005 #6

    George Jones

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    It means transpose. For example (5,2,3) is a row matrix (i.e., a 1x3 matrix) and (5,2,3)^t is a column matrix (i.e., a 3x1 matrix).

    Regards,
    George
     
  8. Dec 2, 2005 #7

    JasonRox

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    I know what transpose means. :tongue2:
     
  9. Dec 2, 2005 #8

    HallsofIvy

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    Then I'm surprised that you don't know that AT is a standard notation for transpose.
     
  10. Dec 2, 2005 #9

    JasonRox

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    Exactly... capital T. :tongue2:
     
  11. Dec 2, 2005 #10

    matt grime

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    You can't guess that lower and upper case t in an expression x^t or x^T where x is a vector don't both obviously mean transpose?
     
  12. Dec 2, 2005 #11

    JasonRox

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    I didn't think the transpose was necessary.

    Anyways, the question isn't about transposes.
     
  13. Dec 2, 2005 #12

    matt grime

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    It's about vectors, and linear maps. What on Earth was it going to be except transpose?
     
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