[Linear Algebra] quick question

[iCeCuBEz v2]

the answer to this seems really obvious but I just don't understand what this question is asking. I know it involves the definition of linear transformations but I'm confused as to what it is asking.

1.) Let C1 be the space of all differentiable functions.
a) Using the definition of linear transformation on page 204, prove that
T : C1 -> C1 defined by T(f) = f' for all functions f is an element of C1 is a linear transformation.

[emerald000]

A linear transformation is a function between 2 modules that keep intact the addition and scalar multiplication.

It is asking you to prove that this definition applies to derivatives:

f_1 + f_2 = f'_1 + f'_2
and
kf = kf'

This should be easy to prove.

[smartdude1212]

Yeah, just need some basic properties from calculus about differentiation here along with your two requirements for linear transformations.

[reuben_tate]

Remember to show that T(f) is a linear transformation when T is defined as some mapping from A -> B and f is an element in A, we need to show the following:

T(a+b) = T(a) + T(b) for all a,b in A
and
T(k*a) = k*T(a) for all a in A where k is a scalar multiple

As long as you know your basic rules regarding differentiation, this should be easy to show if T is defined as the differentiation operator.

Edit:
Might give it away but if you are still stuck:

How does one differentiate x^2 + x? Well, we can get that by differentiating the two separatly and then adding the result. let me define D(f) as the derivative of f. Then symbolically we have:
D(x^2+x) = D(x^2) + D(x) = (2x)+(1) = 2x+1

What about 2*x^2? Well that's the same as finding the result of diff x^2 and multiplying by 2:
D(2*x^2) = 2*D(x^2) = 2*(2x) = 4x

I'm hoping thesd examples give you a clue of what to do.

This was hard to type on my phone btw xP

[iCeCuBEz v2]

omg im retarded I knew about the requirements for a linear transformation and what derivatives are as well I just couldn't figure out what the question was asking XD

thanks.

[reuben_tate]

Originally Posted by iCeCuBEz v2

omg im retarded I knew about the requirements for a linear transformation and what derivatives are as well I just couldn't figure out what the question was asking XD

thanks.

When I took Lin Alg I remember my professor saying that knowing and understanding definitions is very crucial to doing well in the course. My advice is to follow that advice: you can't know what to do if you don't understand what the problem is asking of you. :P