Non-linear Lines . How to Avoid Lying, without Confusing.

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Main Question or Discussion Point

"Non-linear Lines". How to Avoid Lying, without Confusing.

Hi, everyone:

I will be teaching an intro course in Linear Algebra this Spring.

Problem I am having is that the definition of linear does not

apply to lines that do not go through the origin:

Let L:x-->ax+b

Then L(x+y)=ax+ay+b =/ L(x)+L(y)

similarly: L(cx)=acx+b =/ c(L(x))=cax+cb

Which is true only for c=0 . So lines are affine objects, carelessly described as linear,
as in 'linear equations'

So, how does one reasonably avoid bringing up the issue of affine vs. linear
and still not refer to a collection of equations

ax_i +b=0

as linear equations?
 

Answers and Replies

  • #2
Fredrik
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Is this really a problem? How about telling them something like this: Mathematicians have found it useful to have a term for maps that satisfy T(ax+by)=aTx+bTy. You (the students) will understand why before this course is over. The term they chose for such maps is "linear". It would make just as much sense to use that term for maps that satisfy T(ax+by)=aTx+bTy+z for some z, because all such maps take straight lines to straight lines. The problem is, if we call the members of this larger set of maps "linear", then what do we call "linear maps with z=0"? "Linear and origin-preserving"? This gets annoying pretty quickly, so we just call them "linear".


Student: OK, but then what do we call the maps with arbitrary z?

Teacher: **** you!
 
  • #3
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I'm glad someone is giving this topic consideration.

Let us generalise (as mathematicians are wont to do).

Thre are many examples where less than perfect terminology has arisen, become entrenched and become the source of confusion.
All too often this is hidden by promoting a simplified definition, hammered into earlier years students, who then find it difficult to 'unlearn' the half truth in order to embrace the wider picture.

Why not simply be up front about this and explain that there is some unfortunate terminology so careful attention needs to be placed upon the boundaries within which your definitions apply.

In all conscience students who go on to greater things will meet plenty of such examples in their future studies. The rest need not worry, but simply stay within the guidelines.
 
  • #4
atyy
Science Advisor
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Just say there are (at least) two different definitions of "linear".

Even the OED gives multiple definitions of words.

And that in this course, we use this particular definition throughout.
 

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