# Determination of Linear transformation

1. Nov 16, 2008

### 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 )

2. Nov 16, 2008

### Redbelly98

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

3. Nov 16, 2008

### Maxwhale

(rx, ry)

4. Nov 16, 2008

### Redbelly98

Staff Emeritus
Yes.

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

5. Nov 16, 2008

### Maxwhale

yeah i have done that

6. Nov 16, 2008

### Redbelly98

Staff Emeritus
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:

Last edited: Nov 16, 2008
7. Nov 16, 2008

### Maxwhale

should it be

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

8. Nov 16, 2008

### Redbelly98

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

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