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Linear Transformation

  • Thread starter sana2476
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  • #1
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Homework Statement



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)?



The Attempt at a Solution



is it safe to say that identity matrix forms a basis? I need help understanding this problem
 

Answers and Replies

  • #2
dx
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What are the components of α+βτ+γτ² in the basis (1,τ,τ²)?
 
  • #3
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The components would just be α=1, β=1, γ=1. Isn't that right?
 
  • #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)?
 
  • #5
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a,b,c would be the components.
 
  • #6
dx
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Yes, that's correct. Now what are the components of α1ττ² in the basis (1, τ, τ²)?
 
  • #7
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α,β,γ are the components in the basis (1, τ, τ²).
 
  • #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)?
 
  • #9
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you would get:

[2α + γ]
[β + 3γ]

Correct?
 
  • #10
dx
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Yes, you get the vector (2α + γ, β + 3γ). But what basis is this vector in?
 
  • #11
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It's in the basis: (1, τ, τ²)?
 
  • #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?
 
  • #13
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The basis for B is (1,τ)
 
  • #14
dx
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Ok, so what's the answer? What's L(α+βτ+γτ²)?
 
  • #15
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Ok so (1,τ,τ2) is the basis for L(α+βτ+γτ²)
 
  • #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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