Question

6) Show that if L is a linear operator, ie. L(g/t Gf,)-G L(J) +GJ (G) , then 1 (gf, + c2f2 + c3f3)-Q. (f) + GL(7) + c,L(左) Hint: you might find it easier to begin with the right hand side of this expression. You don't have to prove this but it should be obvious after you're done that this extends inductively to a linear combination of any number of functions

Answer 1

Show that if L is a linear operator i.e L (c_1 F_1 + c_2 F_2) = c_1 L (F_1) + c_2 L (F_2), then L (c_1 F_1 + c_2 F_2 + c_3 F_3) = c_1 L (F_1) + c_2 L (F_2) + c_3 L (F_3) Rightarrow Let given that L is linear operator and let F_1, F_2, F_3 are functions and c_1, c_2 and c_3 are scalars, then L [c_1 F_1 + c_2 F_2 + c_3 F_3] = L [(c_1 F_1 + c_2 F_2) + c_3 F_3] leftarrow adjacent = L (c_1 F_1 + c_2 F_2) + L (c_3 F_3) leftarrow Because L is linear operator L (c_1 f_1 + c_2 f_2 + c_3 f_3) = c_1 L (f_1) + c_2 L (F_2) + c_3 L (f_3) leftarrow Because again applied L is linear operator