Show that the function T R2 rightarrow R2 given by Txy yx i

Show that the function T: R^2 rightarrow R^2 given by T(x,y) = (-y,x) is a linear transformation.

Proof. We need to show that if a = (a1, a2) and b = (b1, b2)

then T(a + b) = T(a) + T(b)

and for a scalar k we have T(ka) = kT(a).

T(a + b) = T((a1, a2) + (b1, b2))

= T(-a2-b2, a1+b1)

=T(a1, a2) + T(b1, b2)

= T(a) + T(b)

We also have T(ka) = T(k(a1, a2)

= T(ka1, ka2) =

=<-ka2-kb2,ka1+kb1>

= k<-a2-b2,a1+b1>

Therefore, T is a linear transformation.