Prove that T is a linear operator

Question

Let T:R²→R² be the function defined by T(〈x,y〉) = 〈-y,x〉.

Hint: We want to show T(u + v) = T(u) + T(v) and T(λu) = λT(u) For all u,v ∈ R² and λ ∈ R.

Answer 1

u = <x,y>

v = <z,w>

T(u + v) = T(<x + z, y + w>) = <-(y + w), x + z> = <-y,x> + <-w,z> = T(u) + T(v)

Good.

T(λu) = T(λ<x,y>) =T(<λx,λy>) = <-λy,λx> = λ<-y,x> = λT(u)

Good.

T is a linear operator over R².

Answer 2

The notation used in the formulation of this question is a bit strange. Normally even if you use <x,y> to denote a vector, a function from R² to R² can just be expressed as f(x,y). So the question is how to prove that the function f define by f(x,y)=(−y,x) is linear.

If we write u=(x,y), v=(x',y'). Then u+v=(x+x',y+y'), and

f(u+v)=f(x+x',y+y')=(−(y+y'),x+x')=(−y−y',x+x')=(−y,x)+(−y',x')=f(x,y)+f(x',y')=f(u)+f(v).

For a scalar λ∈R, λu=λ(x,y)=(λx,λy). Thus f(λu)=f(λx,λy)=(-(λy),λx)=(λ(−y),λx)=λ(−y,x)=λf(u).

Therefore, f is linear.