Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive (i. e. satisfies f(v+w) = f(v)+f(w) for v,w in C) but not linear as a map of complex vector spaces. Show however that if we define f as the map f : R^2 -> R^2 given by f(x,y) = (x+2y, 3x+4y) then f is linear as a map of real vector spaces.

Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive (i. e. satisfies f(v+w) = f(v)+f(w) for v,w in C) but not linear as a map of complex vector spaces. Show however that if we define f as the map f : R^2 -> R^2 given by f(x,y) = (x+2y, 3x+4y) then f is linear as a map of real vector spaces.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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