Linear Transformations (again)

V is the vector space of functions that have continuous 2nd derivatives on (2, 5). (meaning y' and y are both also continuous on (2, 5)). W is the vector space of continuous functions on (2, 5).

a) L is the function that assigns to each such "input" function y the "output" function y'y. For each input function y, the output function is called L(y). Then L(y)=y'y, which means that for every t allrealnumbers (2, 5), L[y](t) = y'(t)y(t). For example, if y(t)=t^3 then L[y](t) = (3t^3)(t^3)

b) L is the function that assigns to each such "input" function y the "output" function 4y'' - 3y' + 2y. For each input function y, the output function is called L[y]. Then L[y] = 4y'' - 3y' + 2y, which means that for every t allrealnumbers (2, 5), L[y](t) = 4y''(t) -3y'(t) + 2y(t)