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)
I apologize. The question is the same as the other post I made. By checking both i and ii (as follows), determine whether of not the function L is really a linear transformation.
vectors denoted by bold
i) for every m and n in V, L(m+n)=L(m) + L(n)
ii) for every m in V and every scalar c, L(cm) = c(L(m))
Okay, for the first one, the question can be answered by looking, as I suggested, at and . Are they the same?
For the second one, you have to be more general. Look at where a and b are numbers, f and g are twice differentiable functions. L(af(x)+ bg(x))= 4(af(x)+ bg(x))"- 3(af(x)+ bg(x))'+ 2(af(x)+ bg(x)). Is that equal to aL(f(x))+ bL(g(x))= a(4f''(x)- 3f'(x)+ 2f(x))+ b(4g''(x)- 3g'(x)+ 2g(x))?
(My saying that you can use a single example for the first but that the second requires you to be "more general" sort of gives the answer away!)