Show that the T is a linear transformation

[Dank2]

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)).

 

[Ray Vickson]

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?

 

[Dank2]

So, what is your question for us?

Title, to show that T is linear transformation, what i did is enough?

 

[Ray Vickson]

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.

 

[Dank2]

These are the two properties that is required, but the way that it's shown, since it's with a function, is correct?

 

[Mark44]

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?

 

[Dank2]

What are R₂[x] and R₄[x]?
Are these polynomials with real coefficients of degree less than 2 and 4, respectively?

that's correct.

 

[Dank2]

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?

 

[Mark44]

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)]?

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?

 

[Dank2]

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.

 

[Mark44]

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].