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