Calculus How To

Linear Function: Simple Definition, Example, Limit

Contents:

  1. What is a Linear Function?
  2. Linear Combination
  3. Linear Relationships / Non Linear
  4. Linear Equation
  5. Linear Map

1. What is a Linear Function?

linear function

The graph of a linear function has a straight line.




Linear functions are functions that produce a straight line graph.

The equation for a linear function is:
y = mx + b,
Where:

  • m = the slope ,
  • x = the input variable (the “x” always has an exponent of 1, so these functions are always first degree polynomial.).
  • b = where the line intersects the y-axis.

The equation, written in this way, is called the slope-intercept form.

Examples of linear functions:

  • f(x) = x,
  • f(x) = 2x – 2,
  • f(x) = x + 1.


Limit of a Linear Function

You can find the limit of a linear function in several ways, including:

A More Formal Approach

You can’t always use a table or calculator in class; Sometimes you have to make a more formal approach, using the definition of a limit. The following example shows how to do this for the function y = 2x + 2. Slightly different steps are needed to solve for infinity, and values other than infinity.

Solving for limits of linear functions approaching values other than infinity.

Example problem: Find the limit of y = 2x + 2 as x tends to 0.

The limit for this function is 0 at x = 0, and ∞ for x=∞


Step 1: Set up an equation for the problem:Use the usual form for a limit, with c equal to 0, and f(x) equal to 2x + 2.
f(x) = 2 x + 2
c = 0
lim f(x) = L = lim 2x + 2
x→c x→0

Step 2: Solve for the limit of the function, using some basic properties of linear functions:

  1. The limit of ax as x tends to c is equal to ac
  2. The limit of a as x tends to c is a
  3. The limit of a + b is equal to the limit of a plus the limit of b

Using this logic, the limit is 2 as x approaches 0.
lim(x→0) 2x + 2 = lim(x→0) 2x + lim(x→0) 2 = 0 + 2 = 2

Solving for limits of linear functions approaching infinity.

Example problem: Find the limit of 2x + 2 as x tends to 0.

Step 1: Repeat the steps as above, but this time solve for the limit as x approaches infinity.
f(x) = 2x + 2
c = ∞
lim(x→&infin) 2x + 2 = lim(x→&infin) 2x + lim(x→&infin) 2 = ∞ = Limit does not exist

Tip: Since the limit goes to infinity when you times infinity by 2, the limit of the function does not exist due to infinity not being a real number. There is one special case where a limit of a linear function can have its limit at infinity taken: y = 0x + b. Since the 0 negates the infinity, the line has a constant limit. This would appear as a horizontal line on the graph.
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2. What is a Linear Combination?

In general, a linear combination of a set of terms is where terms are first multiplied by a constant, then added together.

For example, let’s say you have two terms x and y. You might multiply x by 10, and y by 8, to get: 10x + 8y. The expression 10x + 8y is called a linear combination.

The constants placed in front of the terms (10 and 8 in this example) are sometimes called coefficients.

Linear combinations are used frequently because they are easier to conceptualize than some of the more complicated expressions (like those involving division or exponents). Compared to their more complicated relatives, they are also easier to work with mathematically.

Examples of Linear Combinations

Coefficients in a linear combination can be positive, negative or zero. You can also have one term, or more. For example, all of these expressions are valid linear combinations:

  • 10x – 8y + 8z
  • 10y + 8z
  • x + y
  • xz
  • 9y

Linear Combination of Vectors

The above definition also extends to vectors. Let’s say that you have two vectors v and w; Each vector is multiplied by a scalar a and b, giving the expression:
av + bw.
The expression av + bw is called a linear combination of v and w.

Using a little linear algebra, you can show linear combinations of more complicated vectors. As an example, the vector (7, 11, 15) is a linear combination of the vectors (1, 1, 1) and (1, 2, 3). The first vector (1, 1, 1) is multiplied by the scalar 3, and the second vector (1, 2, 3) is multiplied by the scalar 4. Adding the results from each multiplied vector, you get:
linear combination

If you’re unfamiliar with matrix multiplication and how the following answer was arrived at, watch the following short video:

Examples of Linear Combinations of Vectors

  • 10u – 8v + 8w
  • 10u + 8w
  • u + v
  • uzw
  • 9v

3. Linear Relationships

A linear relationship is where you represent the relationship between variables as a line (the word comes from the Latin linearis, from linea “a line”). If you graph linear line, you’ll see a perfectly straight line with no curves.

linear relationship

The five points plotted here have a linear relationship.


Non linear relationships are (perhaps not surprisingly) everything else. If there’s no straight line, then it’s non linear.

This set of data is an example of a non linear relationship.


Linear Equations and Functions

If a set of data has a linear relationship, you can represent it with a linear equation or function.

In algebra, you probably came across linear equations and the slope formula. The slope formula looks like this:
y = mx + b
Where:

  • “m” = the slope,
  • “x” = input (x-value),
  • “b” = the y-intercept (where the graphed line crosses the vertical axis).
linear equations

Two examples of linear equations, with their derived slopes (left) and y-intercept (right).

Linear functions are similar to linear equations. They are functions that can be represented by a straight line graph. A few examples of linear functions that will give a straight line graph:

  • f(x) = x,
  • f(x) = 2x – 2,
  • f(x) = x + 1

The variables in linear functions have linear relationships.

Positive and Negative Linear Relationships

positive and negative linear relationship


The terms “positive” and “negative” linear relationships refer to the direction the function is traveling.

  • Positive linear relationship: The line travels upwards from left to right.
  • Negative linear relationship. Travels downwards from left to right.

Finding Linear Relationships from Data

The easiest way to visualize a linear relationship or recognize a linear function with a small set of data is to make a scatter plot.

This scatter plot shows a clear linear relationship.

This scatter plot shows a clear linear relationship.




For large sets of data, these are best represented by functions, which you’ll be able to graph on a graphing calculator. You can also plug the numbers into a table on the TI-89 and graph a scatter plot that way.

Collinearity

collinear 1

A, F, and B are collinear.

Definition

A set of points is collinear if you can draw one line through them all. The word literally means “together on a line.” Two points are always collinear: no matter where you draw the two points, you can always draw a straight line between them. A general way to write this is: “Points P1, P2 and P3 are collinear”, which can also be written as “point P1 is collinear with points P2 and P3“.


It goes without saying that points are non-collinear if they do not fall on the same line.

How to Show Points are Collinear

It seems reasonable that if you can draw a line through a set of points, then those points are collinear. The trouble is, those points may not be exactly on the same line. One way to work around this is with the knowledge that the points must satisfy the same linear equation. For example, if you are given the linear equation y = 4x + 16, you know that the points (-4, 0) and (-1, 12) meet the definition because (plugging the x and y values into the equation) we get:

  • 0 = 4 * (-4) + 16
  • 12 = 4 * (-1) + 16.

An Alternate Method

A second way is to find the slope between the points (i.e. the slopes of the line segments between points P1 and P2, and P2 and P3); if the slopes are the same then the points are collinear. For example, the set of points in the image below fit the definition if the slope of line segment A equals the slope of line segment B.


Example question: Do the points P1 = (−4, 0), P2 = (−1, 12) and P3 = (4, 32) show collinearity?

Step 1: Find the slope for the line segment between the first two points using rise-over-run =(y2 − y1) / (x2 − x1) = (12 − 0)/( −1 − (−4) )= 12 / 3 = 4

Step 2: Find the slope for the line segment between the next two points =(y3 − y2)/(x3 − x2) = (32 − 12)/(4 – (-1))= 20/5 = 4.

Step 3: Compare the slopes you calculated in Steps 1 and 2. The two slopes equal 4, so the points do show collinearity.

Linear Equation

linear equation

A linear equation graphs a straight line.

A linear equation graphs to a straight line and is a degree-1 polynomial. In other words, each term is either:

  • A constant, or
  • The product of a constant and one variable.

They can be written in the form:

a1x1 + … anxn + b = 0

Here:

  • x1…, xn are the variables, and there may be any number of them.
  • b, a1, … an are the coefficients; they can be constants or expressions, but their expressions can’t include any variable. Some coefficients may be zero, but not all.

Linear Equation Applications

Linear equations are important in physics and engineering. Some physical processes show a direct linear relationship, and even non linear relationships can often be approximated by systems of linear equations. When possible, we like to estimate with them because they are easy to manipulate and calculate with.

Examples of Linear Equations

The simplest linear equation is the one with one variable:
ax + b = 0.
A little bit of algebraic manipulation makes it clear that the unique solution to this linear equation is always -b/a.

If the linear equation has two variables, they are usually called x and y. Then the equation can be written as

ax + by + c = 0

Two independent linear equations will define these two variables completely. Reducing them down to an x = d, y = e form usually requires a small amount of algebraic multiplication. For example, if you are given the two equations

  • 3x – y = 7
  • 2x + 3y = 1

The first equation can be rearranged as y = 3x – 7.

  • Substitute 3x – 7 for y in the second equation, to get:
    2x + 3(3x-7) = 1.
  • Multiply this out to get:
    2x + 9x -21 = 1,
  • Simplify that to get:
    11x = 22.
  • Dividing both sides by 11, we see x is 2.

Finding y is as simple as plugging x = 2 in our first equation, for 6 – y = 7. So y is -1.

Linear equation vs Linear Function

A linear function also has a straight line graph, and can be described by a linear equation. The two terms are so similar that they are often used interchangeably. For all intents and purposes, they are usually the same thing. That said, there is a tiny difference:

  • A linear function is a type of function and so must follow certain rules to be classified as a “function”. For example, functions can only have one output for each input.
  • On the other hand, equations are just statements that make two things equal, like x = y or 52x = 100. A linear equation also makes two things equal, but produces a straight line. That straight line may, or may not be, a function.

Linear Maps

Contents:

  1. Linear Map Definition
  2. Properties
  3. Examples of a Linear Map

What is a Linear Map?

A linear map is a function from one vector space (the domain) into another (the codomain).

If U is the domain and V is the codomain, we can call our linear transformation T, and define it like this:
T:U → V. If U and V are the same, our linear map is called an endomorphism.

Properties

Every linear function has two special properties. For every u1 and u2 in U

T(u1 + u2) = T(u1) + T(u2).

Also, for all u in U and a in C (i.e., for every constant a)

T(au) = aT(u).

These two properties together, are called linearity. Since linear transformations are operation preserving, we can apply them before or after the addition or scalar multiplication, with no difference in the result.

Examples of a Linear Map

The identity map might be the simplest example of a linear transformation. This is the transformation that maps every point into itself.

The function in the real number space, f(x) = cx, is a linear function. This function can be drawn as a line through the origin.

There are many simple maps that are non linear. For example, for real numbers, the map x: x → x + 1 is non linear. So is the mapping x → x2, also over real numbers.

The following series of three images illustrates the linear function f: R2 → R2 with f(x, y) = (2x, y). The y component of the vector remains the same, while the x component is scaled by two, as shown in the first image. The second image shows the additivity of the linear map: it makes no difference whether two vectors are added and then mapped, or whether they are mapped and then added.

The third image demonstrates the linear transformation is homogeneous. Whether the vector is scaled and then mapped, or mapped and then scaled, the final result will be the same.

linear function map


linear function map example


linear map homogeneity

Linear Function: References

Beezer, Robert A. Linear Transformations. from A First Course in Linear Algebra, version 3.50. Created on December 12, 2015. Retrieved from http://linear.ups.edu/html/section-LT.html on November 9, 2019
Belk, J. Linear Combinations and Span. Retrieved December 1, 2019 from: http://faculty.bard.edu/~belk/math213s14/LinearCombinationsAndSpanRevised.pdf
Larson, R. & Edwards, B. (2009). Calculus. Cengage Learning.
Lian, Bong H. Linear Maps. Retrieved from http://people.brandeis.edu/~lian/TsinghuaMathcamp2014/Chapters8-11.pdf on November 9, 2019
Linear Equations. Chm 621 Curve Fitting. Retrieved from http://www.chem.purdue.edu/courses/chm621/text/curve/matrix/linear/linear.htm on January 5, 2018.
Oldham, K. et al. (2008). An Atlas of Functions: with Equator, the Atlas Function Calculator 2nd Edition. Springer.
Tan, S. Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach. Cengage Learning.
Combinatorics of Fine Geometries.

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