Calculus How To

Linear Form: Definition, Examples

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The term “linear form” has a few different meanings in math and calculus. The exact definition depends on where you are using it:

  1. Algebra & Equations for Lines
  2. Differential Equations
  3. Other Areas of Math

1. Linear Form (Equations for a Line)

An equation is in linear form if it is written as y = mx + b. Informally speaking, that’s an “equation for a line.”

If a line can be represented by the equation Ax + By + C = 0, then the equation is a general (or standard) linear form for for the line [1].

linear form

Graph of the line 3y = 2x + 1. The general linear form is the equation 0 = 2x – 3y + 1.


Non-linear equations can be converted to linear forms. Watch the video, which shows you how:


Can’t see the video? Click here.

Linear forms don’t have to be straight lines though. A homogeneous polynomial (which is a multivariate polynomial with terms of the same degree) of degree 1 is a polynomial linear form. The most well-known example of a homogeneous polynomial is the Pythagorean theorem x2 + y2 = z2. Trigonometric functions can also linear (in the sense of a form), as long as the trig functions are raised no higher than the 1st power [2].

An affine function is sometimes defined as a linear Form plus a number. Generally, when dealing with affine functions, they can be expressed with a slightly different format:
c1x1 + … + cnxn.
Adding a constant to this expression gives the affine function:
c1x1 + … + cnxn + b

Mathematical relationships can also have a linear form. For example, two sets of data might closely follow each other in a linear relationship, detectable with a Correlation Coefficient.


2. Differential Equations in “Linear Form”

If a differential equation is in the form:
linear form differential equation
then it is in linear form.

3. Other Areas of Math

Many other areas of math use the term to describe a specific behavior or format. For example, a linear map for a given vector space E over a field K, f: EK, is called a linear form; The set of all of these linear forms is called the dual space of E [3].

References

[1] Chung, S. Understanding Basic Calculus. Retrieved October 10, 2021 from: http://www.math.nagoya-u.ac.jp/~richard/teaching/f2016/BasicCalculus.pdf
[2] Patel, A. Section 6.2 Trigonometric Equations PART 1. Retrieved October 10, 2021 from: http://apps.lonestar.edu/blogs/apatella/files/2014/08/section-62-2.pdf
[3] 17.1 Hyperplanes and Linear Forms. Retrieved October 10, 2021 from: https://www.cis.upenn.edu/~cis610/geombchap17.pdf


CITE THIS AS:
Stephanie Glen. "Linear Form: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/linear-form/
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