045-Linear-Factories.Rmd

# The Paradigm of the Factory

Factories take inputs and combine them to produce outputs.

Examples of factories:


* The cabinet shop
    - making shelves from plywood, varnish, ...
* Nutrition
    - constraint: only so much you can eat
* The Economy (leading up to differential equations)

Examples to save for later:

* Chemical reactions
* Energy
    - constraint: need to satisfy demand

## Looking at factories from two sides

1. Synthesis - how much output will we get?
    - can be more than one output: e.g. cabinets, pollution, 
    - many inputs: rent, wood, hinges, varnish
2. Analysis - how does the factory work?
    We see the factory consuming such and such inputs and producing such and such an output. From this, can we figure out how the factory works.

    
## Three Tasks about Factories 

1. Learning: We measure explanatory variables 
2. Optimization - given a factory, what inputs should we get to produce the best outputs.
    - outputs
    - constraints

## Representing a factory mathematically

1. Functions
2. Constraints


## The linear factory

1. Linear combinations
2. Why linear?
    * Approximates any function
        - greenhouse area is product of width and depth. For small changes, it's linear.
    * Turns the tasks into arithmetic
    * Provides a fantastic metaphor about which we are highly skilled in reasoning: space and subspaces.
3. Nonlinear examples
    * polaroid filters. Two crossed filters block light. A diagonal filter in the middle allows the light to be transmitted.
    * sounds are linear, perceptions are not:
        perceiving two steady notes as separate and constant until the frequency becomes close enough that they sound like a beat
    * distinguishing conversations in a cocktail party. We don't confuse the sentences spoken simultaneously by two people as one sentence. 
    * stretched springs and deformation. Bending until breaking.