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

## 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 the line [1].

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

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 x^{2} + y^{2} = z^{2}. 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:

c_{1}x_{1} + … + c_{n}x_{n}.

Adding a constant to this expression gives the affine function:

c_{1}x_{1} + … + c_{n}x_{n} **+ 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:

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*: *E* → *K*, 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