Show that the function T R2 rightarrow R2 given by Tx y y

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

Solution

T: R² R² is given by T(x,y) = ( -y,x). Let (a,b), (c,d) be two arbitrary elements of R² and let be an arbitrary scalar. Then T[ (a,b) + (c,d)] = T( a+c, b+d) = (-b-d, a+c)= (-b,a) + (-d,c) = T(a,b) + T(c,d). Thus T is closed under addition. Further, t[ (c,d)] = T(c , d) = (-d, c) = ( -d, c) = T (c,d). Thus T is closed under scalar multiplication. Therefore, T is a linear transformation.