linearity ≅ whole=∑parts?

In general, the main condition or step in proving a set to be a linear space is to show that for any two members f, g of the set under consideration, and for any two reals α,β

a. the linear combination αf + βg is in the set
b. this satisfies closure, from whence follow the remaining axioms 
  if applicable.

As an illusration, consider an operator qcq , L: D → ℝ^ℝ , where D is some set of functions , and ℝ^ℝ is the set of all real valued functions of a real variable.

L is linear, ie, is a linear space if , ∀y∈D, ∀z∈D, ∀α∈ℝ , ∀β∈ℝ

L(αy+βz)= αL(y)+βL(z)

This suggests some meaning. Viz., that operations on / properties of the whole are equal to sums of operations on / props of the parts.

This property appears to be akin to that of self-similarity, suggesting a rather profound meaning for the character of natural organization.

The ubiquity of linearity in the abstract algebra by which we represent natural systems^** (as noted earlier) has such consequences as the integrable character of physical law (discussed elsewhere).

^** if theory were complete then one could say up to an isomorphism, ie, the mathematical formalism is in a one-to-one, onto and domain-covering image of the fields of the physical system being described.

outline

 



 special topics (combinging [Apostol] with others):

 linearity (linear combination linearity and linear independence part 1)

 linearity property

 what does the word linear mean /putting linear in linear algebra


 applications

 linear separability (…) TBD



 linear combination (linear combination linearity and linear independence part 2)


 occurences of linear combination



 linear independence (linear combination linearity and linear independence part 3)



 topics not in [Apostol]


 group theory

 projectors

 closure operators

 tensors - or maybe placed after vectors


 linear algebra glossary






listing of linear algebra topics found in [Apostol]: 


 vectors: vector algebra and vector spaces


 linear spaces = abstract vector spaces

 span

 basis,

 dimension, 


 orthogonalization theorem [Waerden] §9 ; Orthogonality [Apostol I] Ch.12 or 15

 norm, orthonormality , coordinate system

 what about the polar coord. system?






 transformations, linear transformations, linear maps, T:V→V, T:V→W , (matrix algebra|matrices)

 transformations (linear only or in general?) 


 linear transformations are matrices

 examples of linear transformations

 matrices ⇔ linear transformations , 


 why matrices can express transformations

 matrix algebra: typology, multiplication,rank,cofactors,minor,



 reversible transformations = nonsingular transformations




 Determinants

 determinant theorems, properties



 singularity reversibility and determinants

 eigenpolynomials, eigenfunctions and eigenvalues


 characteristic functions and eigenvalues, characteristic equation



 systems of linear equations



 general note on linear algebra


 pedagogy note: linear algebra prerequisite for and segues to abstract algebras like group theory general topology (metric spaces), and i guess manifold theory.

 Applications of linear algebra

 the linear algebra formalism is used extensively in quantum mech and quantum field theory (by way of eigenfunctions, complex linear space, and group theory (eg, symmetry or gauge groups)).

 Just like many natural relationships are observed as (or describable by) differential equations on one hand, or by complex analysis expressions, 


 there are many relationships that are expressable instead only by linear equations (or inequalities, as in the case of linear programming or optimization with constraints)

 A system described by a set of linear equations may be solved 1) using linear algebraic techniques , such as matrix representation and techniques , or what can be called matrix analysis.

 examples of such linear relationships

 occur in macro- and microeconomics, resource planning, expenditure planning, etc. (eg, linear programming problems)




 segue to constraint programming, simplex analysis, linear programming, integer programming, optimization problems , &c.







1) 
solved means finding the unknown function