# Show that the T is a linear transformation

## Homework Statement

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

## 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 Helper
Dearly Missed

## Homework Statement

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

## 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?

So, what is your question for us?
Title, to show that T is linear transformation, what i did is enough?

Ray Vickson
Homework Helper
Dearly Missed
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.

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

Mark44
Mentor

## 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.

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:
Mark44
Mentor
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?

f∈R2[x]

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:
Mark44
Mentor
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].