Determine which of the following maps are linear or not: Solution The above map is not linear. Linear maps should satisfy T(f+g) = T(f) + T(g). Consider the following. Let f = 1 and g = -1. Observe that T(f) = (integral 0 to 1) 1^2 dt = 1 T(g) = (integral 0 to 1) (-1)^2 dt = 1 f+g = 0. T(f+g) = (integral 0 to 1) 0^2 dt = 0 So, T(f+g) T(f) + T(g) which shows that T is not linear..