Determination of Linear transformation

[Maxwhale]

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

Determine if the following T is linear tranformation, and give the domain and range of T:

T(x,y) = (x + y2, \sqrt[3]{xy} )

2. Relevant equations

T ( u + v) = T(u) + T(v)

T(ru) = rT(u)



3. The attempt at a solution
1)
let u = (x1, x2);
T(ru ) = T(rx1, rx2)
T(ru )= r(x + y2) , r(\sqrt[3]{xy} )
T(ru ) = r(x + y2 , \sqrt[3]{xy} )

so it suffices the first condition, right?

2)
let u = (x1, y1) and let v = (y1, y2);
T ( u + v ) = T ( x1 + y1, x2 + y2)
T ( u + v ) = Here I am confused with the term ( x + y2)
T ( u + v )


Any help please !

[Redbelly98]

If u=(x,y), what is ru?

[Maxwhale]

(rx, ry)

[Redbelly98]

Yes.

So T(ru) = T(rx, ry) = ???

[Maxwhale]

yeah i have done that

[Redbelly98]

Okay, but you did it wrong.

What do you get when you substitute rx for x, and ry for y, into
T(x,y) = (x + y2, \sqrt[3]{xy} )

EDIT: FYI this is the part that I'm trying to correct:

3. The attempt at a solution
1)
let u = (x1, x2);
T(ru ) = T(rx1, rx2)
T(ru )= r(x + y2) , r(\sqrt[3]{xy} )
T(ru ) = r(x + y2 , \sqrt[3]{xy} )

so it suffices the first condition, right?

[Maxwhale]

should it be

let u = (x1, x2);
T(ru ) = T(rx1, rx2)
T(ru )= ( (rx1 + (rx2)2), \sqrt[3]{(rx)(ry)} )

[Redbelly98]

Yes. Although x1 is x, and x2 is y, of course.

Next (as you know), you compare that expression with rT(u).