Linear transformations

  • #1
26
3
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).
 

Answers and Replies

  • #2
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
18,180
9,860
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).
:welcome:

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.
 
  • #3
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
18,180
9,860
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)##
 
  • #4
26
3
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
 

Attachments

  • 7DC84EDB-8901-4B53-946A-DACB036018A6.jpeg
    7DC84EDB-8901-4B53-946A-DACB036018A6.jpeg
    31.7 KB · Views: 166
  • #5
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
18,180
9,860
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##.
 
  • #6
26
3
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?
 
  • #7
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
18,180
9,860
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.
 
  • #8
26
3
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?
 
  • #9
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
18,180
9,860
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##.
 
  • #10
26
3
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
 

Related Threads on Linear transformations

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
757
Replies
10
Views
812
Replies
13
Views
29K
Replies
1
Views
877
Top