Talk:Linear function (calculus)

Both contents and title are incorrect now. Please correct it or delete it. Thank-you.

The title of this page is now incorrect. It has NOTHING to do with calculus. This is 8th grade mathematics in the USA, 6-8 in the rest of the world.

I would ask you to type in "linear function" in any search engine. With the exception of the completely erroneous article Linear function in Wikipedia EVERY OTHER reference, image, etc. is for linear function as defined here and not one of them is about calculus.

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The contents of this page are now incorrect. There are now major errors in it. Here are just a few.

1. A polynomial of degree 1 requires that the coefficient on x is non-zero (that is the definition of degree of a polynomial). That is, a polynomial of degree 1 cannot be a constant function.
2. The conditions in #2 and #3 are completely incorrect, e.g., if in #2, B=0, you have a linear equation in one variable x whose solution set is a point (or a vertical line in the plane if you are assuming that your range is R². Neither is a function. Similarly for #3. You need to change the non-zeroes to the y if you are including the possibility of a constant function.)

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Quite often the term linear equation is used interchangeably with linear function. While a linear function written in general form is indeed a linear equation, the opposite is definitely not true. For example, 3x=2 is a linear equation in one variable, whose solution set is a point on the number line. And 2x+3y-4z=5 is a linear equation in three variables whose solution set is a plane in 3D, .... Also, in advanced mathematics, a linear mapping is sometimes referred to as a linear function, but this is not the mainstream definition of linear function. Also, a linear function as defined here does not satisfy the conditions of a linear mapping.

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Further, it should be noted that a horizontal line is usually not considered to be a linear function, but is simply called a constant function (for example y=3). Finally, a vertical line is not a function since it does not pass the vertical line test.

Lfahlberg (talk) 18:51, 19 June 2013 (UTC)

P.S. My major complaint with wikipedia is that your articles are "preaching to the chorus". Math articles are written by mathematicians for mathematicians.

The General Form should be listed first. If you think back to when you were in school, the general form of a linear function (i.e. a linear equation in two unknowns) is the VERY first introduction children have to functions. It is the first form in ANY textbook whether we like it or not. I personally do NOT. I have been complaining for years to the math community that this is very confusing, particularly since it is over-dimensioned and thus not-unique, it does not show the true nature of function (i.e. that you put an x-value in the function machine and a dependent y-value comes out), it does not give any useful information (as does the slope-intercept form). But that is the nature of the world we live in.

The article needs to distinguish very carefully between a linear function and a linear equation. The term "linear equation" is very confusing since the word "linear" ONLY applies to the degree of the variables being one and not to the graph. Everyone (including calculus students) thinks that a system of three linear equations in three unknowns is the intersection of three lines. It is the intersection of three planes! The solution set of a linear equation in three unknowns is a plane in 3D.

Further vector-parametric is the correct terminology and not just parametric. One assumes wikipedia should use correct terminology. (One needs to be able to use vector operations on lines when they do get to calculus! It is very hard for students to understand this relationship (especially if the never see the word mention anywhere and cannot find it even in wikipedia).

Lfahlberg (talk) 19:29, 19 June 2013 (UTC)

I realize that the style of the article is now your wikipedia encyclopedia style. But I would think content is slightly more important. Lfahlberg (talk) 19:29, 19 June 2013 (UTC)

I agree that the content is very important. I will try to be brief:

1. While the name "linear function (calculus)" is slightly odd because this meaning is used outside calculus, "linear function (mathematics)" does not distinguish from the other meaning. At least this is the meaning from calculus, while the other is from linear algebra.
2. A constant function is indeed linear, in the normal sense of the word in calculus, because its graph is a line (and, equivalently, because it can be written in point slope form). This is the usage that is used in every calculus book I have seen. I have now removed the claim about "degree one" that I missed before. (Similarly, every square is also a rectangle.)
3. I believe most students see linear functions first in the slope-intercept form, not in the symmetric form ax+by = c.
4. Stewart's book, and every other calculus book I have seen, speaks of "parametric curves", "parametric form", and "parametrization", not "vector-parametric curves", "vector-parametric form", and "vector-parametrization". Moreover, when the parametrization is written as two separate equations, it is explicitly not in vector form, which would write the parametrization as a single vector equation.
5. The condition in the general form was wrong, I have fixed it. The condition in the parametric form is correct, because so long as $x'\not =0$ then the value of x is not constant and so we get a nonvertical line.

— Carl (CBM · talk) 19:38, 19 June 2013 (UTC)

This article is clearly no longer meant for the reader who doesn't understand, is based on a single reference and is full of opinions and inconsistencies (e.g. a.ne.0 in slope-intercept form determines a non-constant function, etc.). However, yours is the power and the strength, I simply do not care to fight any more. I will not return to wikipedia again and have truly lost my belief in its value. Goodbye. Lfahlberg (talk) 20:21, 19 June 2013 (UTC)

I apologize for missing the superfluous condition $a\not =0$ there. It was difficult to see the content when I was going through converting the HTML to wiki syntax. I tried to indicate in my edit summaries that the article was still in very rough shape, and would need significant attention. There is no reason for you to avoid editing it as well when you see inconsistencies. — Carl (CBM · talk) 20:27, 19 June 2013 (UTC)

Help!

I am planning on adding a section that I alluded to earlier in the article. This will deal with obtaining linear functions from a linear equation. The issue is that a choice must be made of which variable is considered the dependent variable. When the variables are called x and y it is conventional to consider y as the dependent variable and this is what is done in this article. However, if different letters are used there is no convention that I am aware of that can be applied. This makes a difference because the linear functions you get when you choose differently are not the same, they are inverses of each other. For the novice this can be very confusing and the source of significant errors, which is why I would like to include the section. My problem is that I can not find a citation for this material (something that talks about chosing the dependent variable in a non-conventional way) and I am loath to commit the WP:OR sin. If anyone can find something, please let me know. By the way, this problem does not arise when the parametric form of the linear equation is used since both variables are dependent on the parameter and no choice is made. This can be used to naively explain why the parametric form generalizes and the other forms do not. I bring this up because this could be a place to look for the citation I am after. Thanks. Bill Cherowitzo (talk) 16:17, 21 June 2013 (UTC)

I don't know of a citation for that. I can say I was thinking that the "general form" might not be as relevant to this article as it is to linear equation. I think it would be safe to just say that, in an equation in general form, ap + bq = c, if b is nonzero then one can solve for q as a function of p, and if a is nonzero one can solve for p as a function of q. That may finesse the issue. — Carl (CBM · talk) 16:21, 21 June 2013 (UTC)

Under normal circumstances, say if I were writing this article from scratch, I would agree – "general form" would never appear as I would not be talking about equations. But, as I was going through the article, this blurring of the distinction between linear functions and linear equations and the claim that this is intentionally done by some authors, I found to be disturbing. Just removing the offending material would not leave enough for a reasonable article, so I thought that rewriting the content without the forced linking was a better way to go. Your finesse, while certainly to be included, does side-step the real problem which is what to do when both a and b are nonzero. In my mind, this question most clearly shows why the two concepts are different - only one linear function can be associated with a line, but two linear functions can be associated with the equation of a line (most clearly seen from the "general form"). Bill Cherowitzo (talk) 17:08, 21 June 2013 (UTC)

Please go ahead and edit things. I agree with the "if we were writing from scratch", but the article is very new, so don't feel too bound by the current structure. All that I have done, in my mind, is try to take the initial draft and clean it up somewhat so that nobody else nominates it for deletion. There is still a lot that can be done. — Carl (CBM · talk) 18:18, 21 June 2013 (UTC)

IMO, the heading "General form" is misleading and should be replaced by "Implicit linear function". Moreover, this section should be linked to Implicit function. Thus the section should be summarized as "If the coefficient of a variable in a linear equation is not zero, the linear equation defines implicitly this variable as a linear function of the other variables. This function may be made explicit by solving the equation in this variable.

By the way, to much of this article duplicates linear equation, and the other sections need also to be rewritten for focusing on functions instead on equations and/or representation of a line. D.Lazard (talk) 09:45, 22 June 2013 (UTC)

Independent (?) variable

As far as I know, the notion of "dependent" and "independent" variables are defined only for implicitly defined functions. In fact, nor "dependent" not "independent" appear in the page function (mathematics). Therefore, I'll remove these words from this page, except when the linear function is implicitly defined. D.Lazard (talk) 10:01, 22 June 2013 (UTC)

My use of these terms was deliberate, even though they are a bit archaic today. Many modern precalculus and college algebra texts still use them (a quick browse at a local book store confirmed this for me) even when the function is explicitly defined. I was aiming the article at an audience for whom these terms would be meaningful. My goal was to set up a vocabulary which would permit me to talk about the closely related, but distinct concepts of equation and function in terms that the intended audience would understand. I think this is an important point to make, as ignoring it leads to much of the sloppiness of the original article. I do agree that there is still too much material here about linear equations and lines, but I was, as is my usual mode of operation, trying to work within the context of the original article. Bill Cherowitzo (talk) 16:46, 22 June 2013 (UTC)

I do not agree that these terms are archaic. There are clearly important when dealing with implicitly defined functions (implicit functions as well as differential equations, for which the unknown function is a dependent variable). Maybe, it is usual for US teachers to not make a clear distinction between "function" and "equation" (this is suggested by the number of elementary articles for which the distinction is not clear). But the intended audience is not restricted to US students in mathematics. It also contain foreigners and computer scientists and many people for whom these terms do not have any clear meaning. Adding confusion to confusion never make things clearer. D.Lazard (talk) 21:11, 22 June 2013 (UTC)

General and parametric form

I am strongly inclined to entirely remove the sections on general form and parametric form, because these are not about linear functions. Those concepts may be about linear equations, or about lines, but this article should focus just on linear functions. — Carl (CBM · talk) 11:32, 28 June 2013 (UTC)

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