# Proving Linear Transformation of V with sin(x),cos(x) & ex

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• Lauren1234
In summary: 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##.Ah okay. So the only thing I need to do is state at the start the trig functions don’t need including and then just copy the rest of what I’ve done?Ah okay. So the
Lauren1234
TL;DR 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).

Lauren1234 said:
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.

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
PeroK said:
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

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Lauren1234 said:
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
PeroK said:
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?

Lauren1234 said:
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
PeroK said:
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?

Lauren1234 said:
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:

PeroK said:
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##.

PeroK said:
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

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in such a way that the properties of linearity are preserved. This means that the transformation preserves the operations of addition and scalar multiplication.

## 2. How do you prove a linear transformation?

To prove a linear transformation, you must show that the transformation satisfies two properties: additivity and homogeneity. Additivity means that the transformation preserves the operation of addition, while homogeneity means that the transformation preserves scalar multiplication. You can also use a matrix representation to prove a linear transformation.

## 3. What is the role of sin(x), cos(x), and ex in proving linear transformation?

Sin(x), cos(x), and ex are all examples of functions that can be used in a linear transformation. They are commonly used because they are continuous and differentiable, making them easy to work with mathematically. They also have specific properties that make them useful in certain types of transformations.

## 4. Can you provide an example of a linear transformation using sin(x), cos(x), and ex?

One example of a linear transformation using sin(x), cos(x), and ex is the Fourier transform, which maps a function in the time domain to a function in the frequency domain. This transformation uses the properties of sine and cosine functions to decompose a function into its frequency components.

## 5. What are some applications of linear transformations using sin(x), cos(x), and ex?

Linear transformations using sin(x), cos(x), and ex have many applications in fields such as signal processing, image processing, and differential equations. They are also used in physics and engineering to model and analyze systems. Additionally, they have applications in computer graphics and animation, where they are used to transform objects and create visual effects.

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