Question

show that the function f:R² →R² given f(x,y)=(x-y, x+y) is linear.

Answer 1

First

$f(\alpha(x, y)) = f(\alpha x, \alpha y) = (\alpha x - \alpha y, \alpha x + \alpha y)$

$= (\alpha(x-y) , \alpha(x+y)) = \alpha((x-y), (x+y)) = \alpha f(x, y)$

Second

$f(x_1, y_1) + f(x_2, y_2) = (x_1-y_1, x_1+y_1) + (x_2-y_2, x_2+y_2) = (x_1-y_1+x_2-y_2, x_1+y_1+x_2+y_2)$

$= ((x_1+x_2)-(y_1+y_2), (x_1+x_2) + (y_1+y_2)) = f((x_1+x_2), (y_1+y_2))$

So f is a linear transformation