Show that the T is a linear transformation

In summary: You can do this by writing down the equation for T[f] and T[g] in terms of f(x) and g(x), and then showing that these equations hold in each space.In summary, T is a linear transformation that takes a function in R2[x] and transforms it into a function in R4[x], where R6[x] is the image of R2[x].
  • #1
213
4

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)).
 
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  • #2
Dank2 said:

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
Ray Vickson said:
So, what is your question for us?
Title, to show that T is linear transformation, what i did is enough?
 
  • #4
Dank2 said:
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
These are the two properties that is required, but the way that it's shown, since it's with a function, is correct?
 
  • #6
Dank2 said:

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?
 
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  • #7
Mark44 said:
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
Mark44 said:
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?
 
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  • #9
Dank2 said:
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
Mark44 said:
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
Mark44 said:
Who told you that?
class teacher.
 
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  • #11
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].
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving the properties of linearity. In simpler terms, it is a transformation that preserves the structure of a vector space, such as the shape and size of the vectors.

2. How can you show that T is a linear transformation?

To show that T is a linear transformation, we need to prove that it satisfies two properties: additivity and homogeneity. These properties state that for any vectors u and v in the domain of T and any scalar c, T(u + v) = T(u) + T(v) and T(cu) = cT(u). If T satisfies these properties, then it is a linear transformation.

3. Why is it important to prove that T is a linear transformation?

Proving that T is a linear transformation is important because it allows us to apply theorems and properties from linear algebra to the function. This makes it easier to analyze and understand T, as well as make calculations and predictions about its behavior.

4. Can a linear transformation have multiple domains and codomains?

Yes, a linear transformation can have multiple domains and codomains. As long as the properties of additivity and homogeneity are satisfied, a function can be considered a linear transformation regardless of the specific vector spaces it maps between.

5. How does proving that T is a linear transformation help in practical applications?

Proving that T is a linear transformation can be useful in various practical applications, such as computer graphics, economics, and physics. By understanding the properties and behavior of T, we can better model and manipulate real-world phenomena and systems, making it a valuable tool in many scientific fields.

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