# Thread: T: R -> R^2 defined by T(x) = [(sinx) (x)].

1. ## T: R -> R^2 defined by T(x) = [(sinx) (x)].

I need to show that this is not linear. From what I understand, sinx makes this a nonlinear function because it is a periodic function and only defined if: -π/2 ≤x≤π/2. Is my conjecture correct? How can I prove it?

Yvonne

2. Originally Posted by yvonnehr
I need to show that this is not linear. From what I understand, sinx makes this a nonlinear function because it is a periodic function and only defined if: -π/2 ≤x≤π/2. Is my conjecture correct? How can I prove it?

Yvonne
A function f(x) is linear iff
$\displaystyle f(x + y) = f(x) + f(y)$ for all x, y in the domain
and
$\displaystyle f(ax) = a \cdot f(x)$ for any scalar a and x in the domain.

Neither condition is satisfied for your function.

-Dan

3. ## definition of linear transformation

I assumed that this function did not satisfy the conditions stated by the definition. However, I don't know how to show that for the sin function.

4. Originally Posted by yvonnehr
I assumed that this function did not satisfy the conditions stated by the definition. However, I don't know how to show that for the sin function.
Well,
$\displaystyle sin(x + y) = sin(x)~cos(y) + sin(y)~cos(x) \neq sin(x) + sin(y)$

And as a particular example:
$\displaystyle sin(2x) = 2~sin(x)~cos(x) \neq 2~sin(x)$

-Dan