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Topic: Standard form of line
Replies: 1   Last Post: Jun 5, 1996 4:04 PM

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Susan L. Addington

Posts: 32
Registered: 12/6/04
Re: Standard form of line
Posted: Jun 5, 1996 4:04 PM
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Arron and students: Thanks for a good question.

Lou, I disagree. While I agree it's a mistake to concentrate solely on
memorizing standard forms and their names, the concept of a standard (or
"canonical") form is an important one in mathematics.

Example: What does the solution set (graph) of
2x^2 + 6x + 7y^2 -3y + 42 xy + 1/7 = 0
look like?
I don't know the answer. If the equation were in
a different standard form, I could read it right off.
This standard form isn't good for that, and takes lots
of algebraic work to get it into the right form.
This form does neatly collect all the like terms, though.

2 more examples:
2x + 3y = 1 is the equation of a line in the plane;
the vector (2,3) is perpendicular to the line.
This determines the direction of the line; you can also
find a point it goes through, to completely describe it.
2x + 3y + 7z = 42 is the equation of a plane in space;
the vector (2,3,7) is perpendicular to the plane.
This determines the direction of the line; you can also
find a point it goes through, to completely describe it.
(This generalizes to higher dimensions, too.

If these examples are too pedestrian, here's a more esoteric one.
The theory of canonical forms for matrices generalizes
directly to decomposition theorems for Lie groups.

Susan Addington
until June 15, 1996: After June 15, 1996:
The Geometry Center Math Department
1300 South 2nd St., Suite 500 California State University
Minneapolis, MN 55454 San Bernardino, CA 92407
(612) 624-5058 (909) 880-5362
fax: (612) 626-7131 (fax) (909) 880-7119
WWW at Geometry Center:
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Check my Web pages for The Number Bracelets Game.

On Wed, 5 Jun 1996, Lou Talman wrote:

> Arron Eisen asks:

> > My students and I are wondering about the need for the standard form of the
> > line when the form y=mx+b contains more easily accessible information about
> > the nature of the line. So, Why do we need the standard form?
> >

> I think it a mistake to catalog the "forms" into which equations of lines can
> be shoehorned. This is another (of many) strategies for replacing thought
> with rote memorization.
> Students should learn to recognize linear equations as such, regardless of
> what "form" they appear in. Then they should learn to manipulate an equation
> until they can obtain the information they need from it.
> --Lou

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