please write answer by hand Decide without proof which of th

please write answer by hand

Decide (without proof) which of the following maps are linear. f: R^2 rightarrow R^2 with f(x, y) = (cos (alpha) x - sin(alpha)y, sin(alpha)x + cos (alpha)y), with alpha [0, 2 pi). f: R^2 rightarrow R with f (x, y) = (x + 2y) (2x - y). f: R^3 rightarrow R^3 with f (x, y, z) = (sin(2x) - 2 sin(x) cos(x) + y, 0, z). f: R^3 rightarrow R^3 with f(x, y, z) = (x + z, y + z, x + y/z). Show conclusively that the map f: R^2\\{(0, y): y R} rightarrow R defined by f(x, y) = x + y^2/x is not linear. Show conclusively that the map f: R^2 rightarrow R^2 defined by f(x, y) = (y, x - y)is linear.

Solution

Question 1).

a). linear, as x,y are appearing in linear form.

b). non-linear as x,y are not appearing in linear form.

c). non-linear as x,y are not appearing in linear form.

d). non-linear as x,y are not appearing in linear form.