# Determination of Linear transformation

## Homework Statement

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

T(x,y) = (x + y2, $$\sqrt{xy}$$ )

## Homework Equations

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

T(ru) = rT(u)

## The Attempt at a Solution

1)
let u = (x1, x2);
T(ru ) = T(rx1, rx2)
T(ru )= r(x + y2) , r($$\sqrt{xy}$$ )
T(ru ) = r(x + y2 , $$\sqrt{xy}$$ )

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 )

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Redbelly98
Staff Emeritus
Homework Helper
If u=(x,y), what is ru?

(rx, ry)

Redbelly98
Staff Emeritus
Homework Helper
Yes.

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

yeah i have done that

Redbelly98
Staff Emeritus
Homework Helper
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, $$\sqrt{xy}$$ )

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

## The Attempt at a Solution

1)
let u = (x1, x2);
T(ru ) = T(rx1, rx2)
T(ru )= r(x + y2) , r($$\sqrt{xy}$$ )
T(ru ) = r(x + y2 , $$\sqrt{xy}$$ )

so it suffices the first condition, right?

Last edited:
should it be

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

Redbelly98
Staff Emeritus