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Linear Transformation

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Let L : R3[τ] → R2[τ] be a linear transformation, where the bases for the polynomial vector spaces R3[τ] and R2[τ] are (1,τ,τ2) and (1,τ) respectively. We also know the matrix representation for L is:

    A=[2 0 1]
    [0 1 3]

    What is the result of L(α+βτ+γτ2)?



    3. The attempt at a solution

    is it safe to say that identity matrix forms a basis? I need help understanding this problem
     
  2. jcsd
  3. May 5, 2009 #2

    dx

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    What are the components of α+βτ+γτ² in the basis (1,τ,τ²)?
     
  4. May 5, 2009 #3
    The components would just be α=1, β=1, γ=1. Isn't that right?
     
  5. May 5, 2009 #4

    dx

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    No.

    Let's try a more familiar example. What are the components of the vector ai + bj + ck in the basis (i, j, k)?
     
  6. May 5, 2009 #5
    a,b,c would be the components.
     
  7. May 5, 2009 #6

    dx

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    Yes, that's correct. Now what are the components of α1ττ² in the basis (1, τ, τ²)?
     
  8. May 5, 2009 #7
    α,β,γ are the components in the basis (1, τ, τ²).
     
  9. May 5, 2009 #8

    dx

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    Yes. Now, what do you get when the matrix

    [2 0 1]
    [0 1 3]

    acts on the vector (α, β, γ)?
     
  10. May 5, 2009 #9
    you would get:

    [2α + γ]
    [β + 3γ]

    Correct?
     
  11. May 5, 2009 #10

    dx

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    Yes, you get the vector (2α + γ, β + 3γ). But what basis is this vector in?
     
  12. May 5, 2009 #11
    It's in the basis: (1, τ, τ²)?
     
  13. May 5, 2009 #12

    dx

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    No. Read the question again. When you have a linear transformation L : A → B, and you want to represent L by a matrix, you must chose a basis for both A and B. What is the basis of B in this case?
     
  14. May 5, 2009 #13
    The basis for B is (1,τ)
     
  15. May 5, 2009 #14

    dx

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    Ok, so what's the answer? What's L(α+βτ+γτ²)?
     
  16. May 5, 2009 #15
    Ok so (1,τ,τ2) is the basis for L(α+βτ+γτ²)
     
  17. May 5, 2009 #16

    dx

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    I get the feeling you don't completely understand what a basis is. What is a basis?
     
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