functions, equations and linearity
wk 1-3
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:
Homogeneity
f(s,x) = s * f(x)
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:
Zero should transform to zero 2, Negative input -> Output changes the sign
Sum of inputs = sum of outputs
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:
Last updated