Is the derivative a linear transformation on span of exponentials and trig functions?

[Lauren1234]

TL;DR

Linear transformations prove including trig functions

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A).

Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?
(I can do this with numbers but the trig is throwing me).

 

[PeroK]

Summary:: Linear transformations prove including trig functions

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A).

Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?
(I can do this with numbers but the trig is throwing me).

How far can you get before you're stuck?

You might also like to use Latex:

https://www.physicsforums.com/help/latexhelp/

Then you can do things like:

$A = \{e^x, \cos x, \sin x \dots \}$

If you reply you'll see how I typed that.

 

[PeroK]

To show something is a linear transformation, you need to show two things:

1) It's well defined, in the sense that it maps functions (or vectors) to functions (or vectors) of the correct type.

In this case, you have to check that $T$ maps functions in $V$ to functions in $V$. If $T$ maps a function in $V$ to a function outside $V$ then it's not a mapping from $V$ to $V$.

2) That it's linear. Which means that $\forall \ f, g \in V$ and $\lambda \in \mathbb{R}$:

$T(f + g) = Tf + Tg$

$T(\lambda f) = \lambda T(f)$

 

[Lauren1234]

To show something is a linear transformation, you need to show two things:

1) It's well defined, in the sense that it maps functions (or vectors) to functions (or vectors) of the correct type.

In this case, you have to check that $T$ maps functions in $V$ to functions in $V$. If $T$ maps a function in $V$ to a function outside $V$ then it's not a mapping from $V$ to $V$.

2) That it's linear. Which means that $\forall \ f, g \in V$ and $\lambda \in \mathbb{R}$:

$T(f + g) = Tf + Tg$

$T(\lambda f) = \lambda T(f)$

Got you. With mine I am confused as to why I haven’t included the trig functions when proving this. Should I have? Also the bottom bit where I have the trig functions I’m not sure at all what I’m showing there or why I did it

 

[PeroK]

Got you. With mine I am confused as to why I haven’t included the trig functions when proving this. Should I have? Also the bottom bit where I have the trig functions I’m not sure at all what I’m showing there or why I did it

The differential operator is generally linear. That is to say is it a linear operator on the set of all differentiable functions. Although, if you're sharp you may see a subtlety here that is not mentioned by the proof you have quoted.

In any case, if it's linear on all functions then it must be linear on a subset of those functions. I.e. it must be linear on $V$.

 

[Lauren1234]

The differential operator is generally linear. That is to say is it a linear operator on the set of all differentiable functions. Although, if you're sharp you may see a subtlety here that is not mentioned by the proof you have quoted.

In any case, if it's linear on all functions then it must be linear on a subset of those functions. I.e. it must be linear on $V$.

Right so I have to state that at the start so then I don’t need to include the trig functions? And that proves it is a linear transformation? Is what I’ve done a good solution?

 

[PeroK]

Right so I have to state that at the start so then I don’t need to include the trig functions? And that proves it is a linear transformation? Is what I’ve done a good solution?

Well, the solution you posted is clearly out of a textbook of some sort.

 

[Lauren1234]

Well, the solution you posted is clearly out of a textbook of some sort.

Well from a different help forum yeah. But I wanted to fully understand what I was doing and not just copy it down exact. Also from the bottom bit could I use this to verify if one of the functions is contain T(V) or is that something different again?

 

[PeroK]

Well from a different help forum yeah. But I wanted to fully understand what I was doing and not just copy it down exact. Also from the bottom bit could I use this to verify if one of the functions is contain T(V) or is that something different again?

The bottom four lines show that $T$ maps $V$ into/onto $V$. This can also be written as $T(V) = V$. That's just another way of saying what I said in post #3:

In this case, you have to check that $T$ maps functions in $V$ to functions in $V$. If $T$ maps a function in $V$ to a function outside $V$ then it's not a mapping from $V$ to $V$.

 

[Lauren1234]

The bottom four lines show that $T$ maps $V$ into/onto $V$. This can also be written as $T(V) = V$. That's just another way of saying what I said in post #3:

Right ok so that just ties everything together to show why we can use a f(x) and g(x)? Sorry for the questions I’m very new to this and am finding the terms hard to get my head around