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Show that the T is a linear transformation

  • Thread starter Dank2
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  • #1
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Homework Statement



T:R2[x] --> R4[x]
T(f(x)) = (x^3-x)f(x^2)

Homework Equations




The Attempt at a Solution


Let f(x) and g(x) be two functions in R2[x].
T(f(x) + g(x)) = T(f+g(x)) = (x^3-x)(f+g)(x^2) = (x^3-x)f(x^2) + (x^3-x)g(x^2) = T(f(x)) + T(g(x)).
let a be scalar in R:
aT(f(x)) = a(x^3-x)f(x^2) = (x^3-x)af(x^2) = T(af(x)).
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



T:R2[x] --> R4[x]
T(f(x)) = (x^3-x)f(x^2)

Homework Equations




The Attempt at a Solution


Let f(x) and g(x) be two functions in R2[x].
T(f(x) + g(x)) = T(f+g(x)) = (x^3-x)(f+g)(x^2) = (x^3-x)f(x^2) + (x^3-x)g(x^2) = T(f(x)) + T(g(x)).
let a be scalar in R:
aT(f(x)) = a(x^3-x)f(x^2) = (x^3-x)af(x^2) = T(af(x)).
So, what is your question for us?
 
  • #3
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So, what is your question for us?
Title, to show that T is linear transformation, what i did is enough?
 
  • #4
Ray Vickson
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Title, to show that T is linear transformation, what i did is enough?
I suggest you try to answer that for yourself. Ask yourself: have you verified all the required properties that a linear transformation would have? If you are unsure, then ask yourself which properties you think you have missed.
 
  • #5
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These are the two properties that is required, but the way that it's shown, since it's with a function, is correct?
 
  • #6
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Homework Statement



T:R2[x] --> R4[x]
What are R2[x] and R4[x]?
Are these polynomials with real coefficients of degree less than 2 and 4, respectively?
 
  • #7
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What are R2[x] and R4[x]?
Are these polynomials with real coefficients of degree less than 2 and 4, respectively?
that's correct.
 
  • #8
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What are R2[x] and R4[x]?
Are these polynomials with real coefficients of degree less than 2 and 4, respectively?
I have been told now it should be R6[x].

How can i see that it's right?, that's it R6[x] and not R4[x] or i can't see it and it depends on f(x) and it's pre-defined?
 
Last edited:
  • #9
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I have been told now it should be R6[x].
Who told you that?
Since ##f \in R_2[x]##, then f(x) = ax +b, right? According to the formula you posted, what is T[f(x)]?
Dank2 said:
How can i see that it's right?, that's it R6[x] and not R4[x] or i can't see it and it depends on f(x) and it's pre-defined?
 
  • #10
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f∈R2[x]
Forgot about that simple fact.

T(f(x)) = T(ax+b)= (x3-x)*(ax2+b) , which is really a polynomial of degree 5 , so the image of transformation is indeed at R6[x] - All the polynomial of degree 5 or less.3
Who told you that?
class teacher.
 
Last edited:
  • #11
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I got confused since different textbooks describe spaces like R6[x] as "polynomials of degree less than 6" and others describe the as "polynomials of degree less than or equal to 6." Also, the notation I've seen many times is p2[x] to mean the same as your R2[x].

The work you did in the first post seems OK to me, but I think the intent of the problem is that you should show how things work with the specific spaces that are given. It should be easy to show that T[f + g] = T[f] + T[g] and that T[af] = aT[f], but I believe they want you to use generic functions in R2[x].
 

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