Re: Universal definition \ meaning of _linear_


True Raptor wrote:
When encountering the world _linear_ in math literature, does this just
mean whatever is being talked about in the context satisfies the
SUPERPOSITION PRINCIPLE ?

The "superposition principle" is a consequence of linearity and closely
enough related to one definition of linearity that I'm going to assume
you understand that definition. (Feel free to correct me!)

Unfortunately "linear" is a short word, and it is "abused" by having
at least a few related but different mathematical meanings.

One you will be familiar with from pre-college algebra. A polynomial
is often called linear to mean that it is of degree one. When a
mathematician wants to be able to distinguish between functions
that are "linear" in the sense that coordinates with an appropriate
superposition principle, the term "affine" will be used to describe
functions that are like polynomials of degree one, ie. a truly linear
function plus a (possibly nonzero) constant.

Another sense of linear is "one dimensional", though this is often
used in common parlance rather than advanced math.

The phrase "linear space" is often synonomous with vector space,
but in combinatorics it means something quite different:

http://planetmath.org/encyclopedia/LinearSpace2.html

(though I suppose the two notions are closely enough related
to be disturbing!).

regards, chip

.

Relevant Pages

  • Universal definition meaning of _linear_
    ... When encountering the world _linear_ in math literature, does this just mean whatever is being talked about in the context satisfies the SUPERPOSITION PRINCIPLE? ...
    (sci.math)
  • Re: cubes mod m
    ... remember that quasi proved linear factors ... will cause all moduli classes to be represented in the primes? ... the collection of polynomials degrees allowed is actually ... and g represents factors equiv m2 ...
    (sci.math)
  • Re: solving the equation
    ...    to have to deal with? ... They are not necessarily linear functions. ... independent functions, which means that no linear combination ... An example set might be the polynomials: ...
    (sci.math)
  • Re: solving the equation
    ... your's neatly with out any confusion. ... They are not necessarily linear functions. ... An example set might be the polynomials: ... Ans c) completion of my problem statement clearly...? ...
    (sci.math)
  • Re: Symmetrized complex space
    ... what (linear) "foldings" ... generated by reflections. ... I will escribe this one quotient. ...
    (sci.math)