# Writing A Map As A Composition of Three Linear/Bilinear Maps

• azdang
In summary, the conversation discusses how to write the map F as a composition of three linear or bilinear maps and then use the chain rule and generalized product rule to compute DF(\phi). There is confusion about what the three maps should be, with suggestions including the map for \phi, a map for \phi\phi, and a more general map such as I. The conversation also mentions that \int f(t)dt is a linear function and f(x,y)= xy is multi-linear. There is uncertainty about which maps to use and how they should be composed.
azdang

## Homework Statement

Consider the map F: C0([a,b],Rn) --> R, F($$\phi$$)=$$\int\phi(t)\phi(t)$$dt (This is the integral from a to b). Write F as a composition of three maps, each of which is linear or bilinear. Then use the chain rule and generalized product rule to compute DF($$\phi$$).

## The Attempt at a Solution

The only maps I could think of would be something like this:

$$\phi$$: [a,b] --> Rn

A: C0([a,b],Rn) --> C0([a,b],Rn)

And then maybe I would need a map for $$\phi$$$$\phi$$? But I'm not sure what that would be or if that's even correct. I'm not even sure that I am on the right track at all with the maps. Does anyone have any ideas?

Is it possible that F is one of the three maps? So that the composition becomes F of A of phi?

Of course, $\int f(t)dt$ is a linear function, from the set of integrable functions to the set of real numbers, itself. Are you including that? And f(x,y)= xy is multi-linear.

I'm not sure I know what you mean.

Isn't F already that map of integrable functions? Or will I need a more general one such as I: C0([a,b],Rn) --> Rn as one of my maps?

Then, what I'm thinking is that it should be:

I of 'something' of phi. I can't figure out what that middle map would be though.

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## 1. What is the purpose of writing a map as a composition of three linear/bilinear maps?

The purpose of writing a map as a composition of three linear/bilinear maps is to break down a complex map into simpler components that can be easily understood and manipulated. This technique is commonly used in mathematics and physics to solve problems involving multiple variables.

## 2. How do you determine the order in which the maps are composed?

The order of composition is determined by the direction of the map. In other words, the first map in the composition is the one that is applied first, followed by the second map, and then the third map. This order is crucial as it can affect the final result.

## 3. What are the properties of a linear/bilinear map?

A linear map is a function that preserves addition and scalar multiplication, while a bilinear map is a function that is linear in each of its arguments. This means that the map follows specific rules when manipulating inputs and outputs, making it easier to analyze and work with.

## 4. Can a composition of three linear/bilinear maps be simplified further?

Yes, in some cases, a composition of three linear/bilinear maps can be simplified further by combining the individual maps into a single map. This simplification can make the map easier to work with and can also reveal hidden relationships between the different maps.

## 5. What are the applications of writing a map as a composition of three linear/bilinear maps?

This technique is commonly used in various fields, such as physics, engineering, and computer science, to solve problems involving multiple variables or complex systems. It can also be used to study and understand the behavior of linear/bilinear maps and their relationships with other mathematical concepts.

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