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description: wk 1-3
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# functions, equations and linearity

Expressions vs Equations

* Expression: Combination of variables
* Expression with a Name = Function, as long as it is single valued.
* Equation: Expression = Constant  \
  -mx + y = c  \
  -> usually have equal sign/equality on the entire equation

A function is a transformation of the input into the output; where input can be vectors and matrices.



Linearity

* Transformation of a single, real value.
* Must meet following two conditions:

1. Homogeneity

* f(s,x) = s \* f(x)

2. Additivity

* Sum of two inputs is sum of each of the inputs added by each other. (ie. F(a+b) = F(a) + F(b)



Tests:

1. Zero should transform to zero   \
   2, Negative input -> Output changes the sign
2. Sum of inputs = sum of outputs
3. Input scaled, output scales. Not vice versa



**Linearity of Multivariable Functions**

A function with multiple input variables (group of number WITH a specified order) is a vector\
ie. f(x,y)

* Has n dimensions (x element of Real number ^n)

Once again must:\
-> must fulfill homogeneity via the same scaling factor s in f (sx, sy)



Proof shown in textbook, do a copy. ????



Matrix is a two dimensional table of numbers that is a product of two vectors

* How to calculate? -> sum of the products of the corresponding elements of each vector.



Function as a Linear Transformation/Mapping\
-> its like saying for each x, we do something on it, and get y.\
-> its like a lambda function

-> Eg. ax = b where a,b element of Real number.

* Every Linear transformation has a unique matrix associated with it.
* Resulting Matrices represents a mapping of n input dimension and m result (which stores the transformed values).



wk 2:

