Linear Transformations (again)

May 2010
4
0
chapel hill, nc
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)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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 don't see a question here! Are you asking which of these is a linear transformation? Is \(\displaystyle L(x^2+ x^3)= L(x^2)+ L(x^3)\) in each case?
 
May 2010
4
0
chapel hill, nc
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))
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Okay, for the first one, the question can be answered by looking, as I suggested, at \(\displaystyle L(x^3+ x^2)\) and \(\displaystyle L(x^3)+ L(x^2)\). Are they the same?

For the second one, you have to be more general. Look at \(\displaystyle L(af(x)+ bg(x)\) 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!)
 
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