Something About Linear Functions


Notation:
- R set of scalars ( e.g. ral numbers )
- α, β, γ, δ, ... scalars ( e.g. real numbers )
- a, b, c, d, ... vectors
- A, B, C, D, ... functions
- ( A o B )( a ) = A( B( a ) ) composition of two function

#Definition
The vectors a and b are linearly dependent
if there exist α and β which not zero, such as that
α * a + β * b = 0.

#Definition
The vectors a and b are said to be linear independet
if they are not linearly dependent.


Definitions above can be easily extent on more then two vectors.

#Definition
Function L is linear if satisfies next two conditions:
- Homogeneity: L( α * a ) = α * L( a )
- Additivity: L( a + b ) = L( a ) + L( b )

Short version:
- Linearity: L( α * a + β * b ) = α * L( a )+ β * L( b ).


Theorem
Composition of linear function is linear function.
Proof :
- Let A and B be a linear function

( A o B )( α * a + β * b ) =
    = A( B( α * a + β * b) )
    = A( B( α * a )+ B( β * b ) )
    = A( α * B( a ) + β * B( b ) )
    = A( α * B( a ) ) + A( β * B( b ) )
    = α * A( B( b ) ) + β * A( B( b ) )
    = α *( A o B)( b ) + β * ( A o B )( b )
QED


Consequence:
Linear function transforms straight line to straight line.
Proof:
- Let L be linear function.
- Formula of straight line is: p = λ * d + s
- - λ is real number
- - d is nonzero vector

L( p ) = L( λ * d + s ) = L( λ * d ) + s = λ * L( d ) + L( s )
QED


Consequence:
Linear function transforms plane to plane.
- Let L be linear function.
- Formula of plane is : p = α * u + β * v
- - α and β are real numbers
- - u and v are linearly independent.
Proof:
L( p ) = L( α * u + β * v )
       = α * L( u ) + β * L( v )
QED