# Linear transformations

• I
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).

PeroK
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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
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Gold Member
2020 Award
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 im 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

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PeroK
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2020 Award
Got you. With mine im 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
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2020 Award
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 text book of some sort.

Lauren1234
Well, the solution you posted is clearly out of a text book 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