Re: Modeling data...Linear or direct proportional

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Dave Simmon asked:
> .... My question is, what do I correctly call the relationship between
> F and 'a' as graphed. Is just "linear" the right term to use? Is there
> another name I should use?

To which Ludwik Kowalski answered:
> I usually reserve the word "proportional" to y=m*x and use the word
> "linear" when y=m*x+b. To make it more acceptable to all I also say
> that "if doubling and tripling of x also doubles and triples y, then the
> relation is proportional.

Technically, the relationship y = m*x is called *linear* because the
mapping preserves linear combinations, i.e. m*(c_1*x_1 + c_2*x_2) =
= c_1*(m*x_1) + c_2*(m*x_2).  The shifted transformation y = m*x + b is
technically called *affine* (being a composition of a linear
transformation and a translation).  In this usage of the term the idea of
'generically makes a geometric line when its range is plotted versus its
domain' is *not* the denotation of the word 'linear' although it is quite
natural to want to give that connotation to that term.

Sam Held commented:
>        You are correct in that y=m*x+b has a linear relation between y
> and x but is not proportional to because if you double x, you get twice
> y plus b.  In strict semantics they are not equivalent.  However, I can
> see people using them interchanging them like velocity and speed.

Yeah, this is probably the best attitude for this situation.  Being pedantic
and saying that the equation for an offset line is not linear, but affine
would probably be more confusing to introductory students than is necessary.
This is a case where maybe fuzzy terminology can be tolerated with little
down side risk.  The students would probably not encounter the pedantic
distinctions between these concepts (linearity, affinity & direct
proportionality) unless and until they take some advanced math courses.

Nevertheless it might be profitable to point out to even *some* beginning
students that calling the form b + m*x 'linear' uses the term 'linear' in
a vernacular way to mean 'makes a line when graphed', and that is *not*
the usual mathematics meaning. The technical mathematics meaning of the
notion of 'linearity' is closer to the conventional idea of direct
proportionality (at least for 1-dimensional transformations that use the
field of ordinary real or complex numbers) than anything directly related
to the generic appearance of a line when the relationship is graphed.

David Bowman
dbowman@georgetowncollege.edu