# Show that the T is a linear transformation

1. May 31, 2016

### Dank2

1. The problem statement, all variables and given/known data

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

3. 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)).

2. May 31, 2016

### Ray Vickson

So, what is your question for us?

3. May 31, 2016

### Dank2

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

4. May 31, 2016

### Ray Vickson

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. May 31, 2016

### Dank2

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

6. Jun 1, 2016

### Staff: Mentor

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

7. Jun 4, 2016

### Dank2

that's correct.

8. Jun 4, 2016

### Dank2

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: Jun 4, 2016
9. Jun 4, 2016

### Staff: Mentor

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)]?

10. Jun 4, 2016

### Dank2

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
class teacher.

Last edited: Jun 4, 2016
11. Jun 4, 2016

### Staff: 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].