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Last modified on February 22nd, 2024

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Linear pair.

‘Linear’ means ‘arranged in a straight line.’ A linear pair of angles comprises a pair of angles formed by the intersection of two straight. Thus, two angles are said to form a linear pair if they are adjacent (next to each other) and supplementary (measures add up to 180°.)

In the below figure, ∠ABC and ∠CBD form a linear pair of angles.

Linear Pair

In the figure below, ∠MOP and ∠PON, ∠PON and ∠NOQ, ∠POM and ∠MOQ, and ∠MOQ and ∠QON are the linear pairs of angles.

linear pair algebra 1 definition

Thus, the linear pairs share a common arm and a common vertex, and their non-common arms are on opposite sides.

However, all adjacent angles do not form linear pairs.

linear pair algebra 1 definition

Here, ∠WYX and ∠WYZ are adjacent angles, but they are not a linear pair.

It states that if two angles form a linear pair, they are supplementary.

Linear Pair Postulate

However, the converse of the above postulate is not true, which means if two angles are supplementary, they are not always a linear pair of angles.

linear pair algebra 1 definition

From (a) and (b), ∠XOZ and ∠PQR are supplementary angles but not linear pairs.

Thus, all non-adjacent supplementary angles are not linear pairs.

If a ray stands on a line, the adjacent angles are supplementary.

Linear Pair Axiom

The converse of the above axiom is also true, which states that if two angles form a linear pair, the non-common arms of both the adjacent angles form a straight line.

Converse of Linear Pair Axiom

Perpendicular Theorem

The linear pair perpendicular theorem states that if two angles of a linear pair are congruent, the lines are perpendicular.

Let us verify this with the following figure, as shown:

Linear Pair Theorem

Here, ∠XOZ and ∠YOZ are congruent angles (m∠XOZ = m∠YOZ).

Since they form a linear pair, we have ∠XOZ + ∠YOZ = 180°

⇒ ∠XOZ + ∠XOZ = 180°

⇒ ∠XOZ = ∠YOZ = 90°

Thus, the lines are perpendicular (OZ ⊥ XY).

Solved Examples

linear pair algebra 1 definition

As we observe, ∠MON and ∠MOP form a linear pair. ∠MON + ∠MOP = 180° ⇒ (x + 1)° + (2x – 70)° = 180° ⇒ (x + 2x)° = 180° + 70° – 1° ⇒ 3x° = 249° ⇒ x° = 83° Thus, ∠MON = (x + 1)° = (83 + 1)° = 84° and ∠MOP = (2x – 70)° = (2 × 83° – 70°) = 96°

linear pair algebra 1 definition

∠ABC + ∠ABD = 180° ⇒ 150° + ∠ABD = 180° ⇒ ∠ABD = 180° – 150° = 30°

linear pair algebra 1 definition

Here, option c) forms a pair of linear.

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Linear pair

A linear pair is a pair of adjacent angles whose non-adjacent sides form a line .

linear pair algebra 1 definition

In the diagram above, ∠ABC and ∠DBC form a linear pair. The angles are adjacent, sharing ray BC, and the non-adjacent rays, BA and BD, lie on line AD.

Since the non-adjacent sides of a linear pair form a line, a linear pair of angles is always supplementary. However, just because two angles are supplementary does not mean they form a linear pair. In the diagram below, ∠ABC and ∠DBE are supplementary since 30°+150°=180°, but they do not form a linear pair since they are not adjacent.

linear pair algebra 1 definition

Linear pairs in polygons

Linear pairs are often used in the study of the exterior angles of polygons :

In a triangle , an exterior angle is the sum of its two remote interior angles.

linear pair algebra 1 definition

For △ABC above, ∠A+∠C+∠ABC=180°. Also, ∠ABC and ∠DBC form a linear pair so,

∠ABC + ∠DBC = 180°

Substituting the second equation into the first equation we get,

∠ABC + ∠DBC = ∠A + ∠C + ∠ABC

Subtracting we have,

∠DBC = ∠A + ∠C

where ∠DBC is an exterior angle of ∠ABC and, ∠A and ∠C are the remote interior angles.

The sum of the exterior angles of any polygon is 360°. This can be shown by using linear pairs. The sum of the interior angles of an n-side polygon is 180(n-2)°. There are n angles in the polygon, so there are n linear pairs. Thus, the sum of the exterior angles is:

180n° - 180(n-2)° = 360°

linear pair algebra 1 definition

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Linear Pair

As mathematicians, we deal with all kinds of pairs. These include ordered pairs , collinear points , inverse numbers, and reciprocals . But what about linear pairs? What exactly is a linear pair, and why is this concept important in the field of mathematics? Let's find out:

What is a linear pair?

A linear pair is formed when two lines intersect, forming two adjacent angles. Here is a picture of ordered pairs:

As you can see, there are a number of ordered pairs in this picture. They include:

  • ∠ 1 and ∠ 2
  • ∠ 2 and ∠ 3
  • ∠ 3 and ∠ 4
  • ∠ 1 and ∠ 4

You might have also noticed that each of these pairs is supplementary, which means that their angles add up to exactly 180 degrees. This also means that linear pairs exist on straight lines.

You should always know that even though all linear pairs are adjacent angles, the same is not true in reverse: Not all adjacent angles are linear pairs.

Topics related to the Linear Pair

Collinear Points

Ordered Pair

Perimeter, Area and Volume

Flashcards covering the Linear Pair

Basic Geometry Flashcards

Common Core: High School - Geometry Flashcards

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Common Core: High School - Geometry Diagnostic Tests

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Linear Pair Definitions and Examples

Linear Pair Definitions, Formulas, & Examples

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Introduction

Linear pair definitions are a powerful way to organize and think about data. By understanding how linear pair definitions work, you can gain a better understanding of your data and how to use it to make better decisions. In this blog post, we will provide you with an introduction to linear pair definitions and show you some examples. We will also explain the benefits of using linear pair definitions in your data analysis, and offer tips on how to get started using them.

Linear Pair of Angles Definition

A linear pair is two angles that have a constant length between them. The most common example of a linear pair is the interior angles of a right triangle, which are 180 degrees. Linear pairs can also be found in other shapes, such as an equilateral triangle or a pentagon.

There are many definitions of linear pair, but the most common is that they are two angles with a constant length between them. There are other definitions that say that they are two angles whose measure is a constant multiple of 180 degrees. The second definition is more general and can be used for any type of angle, not just interior angles.

Another way to think about linear pairs is to imagine them as two lines that intersect at one point. In this context, the length between the lines is called the distance between the points. If you want to find the distance between two points A and B on a coordinate plane, you can use the Pythagorean theorem: AB=AC. This equation tells us that the square of the distance between A and B is equal to the sum of squares of the distances between A and C, B and D, and C and D. So if we wanted to find the distance between A and B on a coordinate plane, we would use: AB=AC+AD+BC=1080+.

Linear pairs can be found in all types ofangles-interior or exterior-and in all positions inside an object. They can also exist outside an object

Properties of Linear Pair of Angles

In mathematics, a linear pair is a pair of angles whose sum is 180 degrees. Linear pairs are important in geometry and trigonometry, as well as other areas of mathematics. A few examples include the following:

A right angle is the simplest example of a linear pair. A straight line can be thought of as a line that goes from the origin (0, 0) to a point on the line perpendicular to the line at the origin. The angle between these two points is 90 degrees.

A 45-degree angle can also be thought of as a linear pair. Imagine taking two semi-circles and placing one half inside the other. Since they have the same radius, their angles are also 45 degrees. If you rotate these semi-circles around their centers so that their newly created angles match those from before, you’ve created a new linear pair that has an angle of 135 degrees.

A 60-degree angle can also be thought of as a linear pair. Imagine taking two lines that intersect at right angles and drawing them to create an L shape. Now imagine cutting off one corner of this L shape so that it’s still an L but with one shorter side (like creating a V). The newly created angle between these two lines is 60 degrees (since it’s halfway between 90 and 180).

Linear Pair of Angles Vs Supplementary Angles

A linear pair is two angles that are related by a right angle. Supplementary angles are angles that are not part of a linear pair. There are six types of supplementary angles: supplementary angle, obtuse supplementary angle, acute supplementary angle, right supplementary angle, inverse supplementary angle, and complementary angle.

Supplementary angles can be created when two straight lines intersect. Obtuse supplementary angle is formed when the lines have a 45 degree intersection and the sum of their slopes is greater than 180 degrees. Acute supplemental angle is formed when the lines have a 30 degree intersection and the sum of their slopes is greater than 90 degrees. Right supplementary angle is formed when the lines have a 15 degree intersection and the sum of their slopes is equal to 90 degrees. Inverse supplementary angle is formed when one line intersects the other at a right angle, forming two acute supplemental angles. Complementary angle is formed when two lines have an equal slope and they intersect at a point that forms an equilateral triangle.

Linear Pair Postulate

The linear pair postulate states that for any two vectors in a Cartesian coordinate system, the two vectors are related by a linear equation. This means that if we know the length of one vector and the length of the other, we can determine their relationship using only math.

In physics and engineering, this theorem is often used to calculate the motion of objects or systems. For example, if we have a set of particles moving in space, we can use the linear pair postulate to determine their movement over time.

What is a Linear Pair of Angles?

A linear pair of angles is a specific type of angle that has two sides that are both linear. In geometry, the slope of a line is the distance from one point on the line to the other divided by the length of the line. This means that a linear pair has two sides with slopes that are equal.

There are many different types of linear pairs, but some examples include:

The total angle formed by two lines is always 360 degrees, no matter how close or far the lines are from each other.

In geometry, it is important to be able to identify all types of angles, since they play an important role in solving problems. Knowing how to create and identify linear pairs can help you solve problems more quickly and accurately.

How Do you Find the Linear Pair of an Angle?

Angles can be measured in degrees, minutes, and seconds. A linear pair is two angles that are equal to each other. To find a linear pair, use the Pythagorean Theorem. The theorem states that in an angle  between two radians counterclockwise from the origin, the length of the hypotenuse is

where A is the length of the shorter side (in meters), and B is the length of the longer side (in meters).

Is a Linear Pair always Supplementary?

A linear pair is a set of two elements that form a rule of association: each element in the first element is associated with an element in the second, and each element in the second is associated with an element in the first. For example, the rule for addition is (1+2) = 3. The three elements 1, 2, and 3 are a linear pair.

There are several types of linear pairs: independent, dependent, associative, and commutative. An independent linear pair has no rules of association between its elements; for example, (x+y)+z = 10. A dependent linear pair has one rule of association: when x and y are paired together, their sum is also paired together (x+y=z). An associative linear pair has three rules of association: when x associates with y, x+y associates with z; when y associates with z, x+y+z associates with 10; and when z associates with 10, y+z associates with zero. A commutative linear pair has two rules of association: when x commutes with y, x+y commutes with z; and when y commutes with z, y+z commutes with 10.

There are several types of Linear Pairs: Independent Linear Pairs have no Rules Of Association Between Their Elements. Dependent Linear Pairs have One Rule Of Association When X And Y Are Paired Together Their Sum Is Also Paired Together

How Many Angles are there in a Linear Pair?

There are three basic types of linear pairs: right angles, acute angles, and obtuse angles. Each type has a specific number of angles in a pair.

Right angles have two angles in a pair, and they are the easiest to identify because their ends are straight. Acute angles have one angle in a pair, and they are typically found near the vertex of an object. Obtuse angles have two angled endpoints, and they are less common than the other two types.

Are Linear Pair of Angles always Congruent?

If two angles in a linear pair are congruent, then the corresponding sides of the triangles are also congruent. In trigonometry, this is called a right angle triangle and the sides are equals in length. The following examples will help you understand how to identify right triangle pairs and determine if they are congruent.

Example: If the angle A is 120 degrees and the angle B is 30 degrees, then their corresponding side lengths would be 6 meters and 3 meters. Because these angles are both 120 degrees, their corresponding sides are also in a right triangle which has a hypotenuse of 12 meters. Therefore, this linear pair is congruent and therefore has a right angle at A-B.

Example: If the angle A is 60 degrees and the angle B is 90 degrees, then their corresponding side lengths would be 2 meters and 1 meter. Because these angles are not both 60 degrees (they are not in a right triangle), their corresponding sides cannot be congruent so this linear pair does not have a right angle at A-B.

Linear pair definitions and examples can be a powerful tool when it comes to studying mathematics. Knowing the different types of linear pair definitions and how they can be used in solving problems can help you become a more proficient mathematician. In this article, we have introduced you to four different types of linear pairs and given you some examples of how each type can be used. Hopefully, this has helped you gain a better understanding of what linear pair definitions are and how they can benefit your learning process.

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Linear Pair of Angles: Definition, Axioms, Examples, Facts, FAQs

What is a linear pair of angles, properties of linear pair of angles, linear pair of angles vs. supplementary angles, solved examples on linear pair of angles, practice problems on linear pair of angles, frequently asked questions on linear pair of angles.

A linear pair of angles is a pair of adjacent angles formed when two lines intersect each other at a single point.  

“Linear” simply means “arranged along a straight line.” We know that a straight angle is an angle that measures $180^\circ$. It is called a straight angle because it appears as a straight line. Two angles formed along a straight line represent a linear pair of angles. 

a straight angle BOA

In the diagram shown below, $\angle POA$ and $\angle POB$ form a linear pair of angles. They add up to $180^\circ$. In other words, they are supplementary.

$\angle POA + \angle POB = 180^\circ$

Observe that these angles have one common arm (OP), which makes them adjacent angles. Also, their non-common sides OA and OB are opposite rays.

Linear Pair of Angles: Definition, Axioms, Examples, Facts, FAQs

Examples of Linear Pair of Angles:

1) In the figure given below, the linear pair of angles are:

$\angle 1 ,\; \angle 4$ 

$\angle 1 ,\; \angle 2$ 

$\angle 3 ,\; \angle 2$

$\angle 3,\; \angle 4$ 

linear pair of angles formed by two intersecting lines

2) Here, $\angle ADC$ and $\angle BDC$ form a linear pair of angles.

example of linear pair of angles

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Definition of Linear Pair of Angles in Geometry

Two angles are said to form a linear pair if

  • they are adjacent angles
  • their non-common arms are opposite rays & form a straight line

Related Worksheets

Count Sides and Angles Worksheet

  • The angles in a linear pair are supplementary (add up to $180^\circ$). 
  • A linear pair of angles are always adjacent angles.
  • A linear pair of angles always form a straight line.
  • They together form a straight angle. 
  • Two angles forming a linear pair have a common vertex and a common arm. Their non-common sides are opposite rays that form a line.

linear pair of angles a and b

Linear Pair Postulate

The linear pair postulate states that if two angles form a linear pair, they are supplementary. 

linear pair postulate

The converse of this postulate is not true. It means that if two angles are supplementary, they do not necessarily form a linear pair of angles.

Counter example: 

In the image below, angles M and N are supplementary since

$120^\circ + 60^\circ = 180^\circ$

However, they do not form a linear pair of angles. They are not adjacent.

angles M and N

Linear Pair Axioms

Let’s learn two important axioms which are collectively termed as ‘linear pair axioms’. Axiom is a mathematical statement that is self-evident and accepted to be true.

Axiom 1: If a ray stands on a line, then the sum of two adjacent angles formed is $180^\circ$ .

The ray OC is standing on a line AB. Thus, the adjacent angles $\angle AOC$ and $\angle BOC$ add up to $180^\circ$.

angles AOC and BOC in a linear pair

The converse of the axiom 1 is also true.

Axiom 2: If two angles form a linear pair, then non-common arms of both the angles form a straight line.

In the diagram shown above, the rays OA and OB are the non-common arms of the angles $\angle AOC$ and $^\circ BOC$. Rays OA and OB form a straight line AB.

Linear Pair Perpendicular Theorem

If two angles forming a linear pair of angles are congruent, then the lines are perpendicular.

linear pair perpendicular theorem

Here, the angles $\angle a$ and $\angle b$ form a linear pair of angles. Also, they are congruent since they measure 90 degrees each.

Thus, the lines x and y are perpendicular to each other.

It is the most common mistake to confuse supplementary angles with linear pairs of angles due to similarity in their properties. However, these are two different terms. Let’s understand the difference.

1. Observe the diagram and identify the linear pair of angles.

lines AB and XY intersecting at point C

The lines AB and XY intersect at a single point C.

Thus, they form linear pairs of angles.

Any two adjacent angles formed here will form a linear pair of angles.

Angles in linear pair are:

  • $\angle ACY$ and $\angle BCY$
  • $\angle ACX$ and $\angle ACY$
  • $\angle ACX$ and $\angle BCX$
  • $\angle BCX$ and $\angle BCY$

2. If two angles forming a linear pair are in the ratio of 7:11, then find the measure of each of the angles.

Solution:  

Let the measures of the angles be $(7x)^\circ$ and $(11x)^\circ$.

Since the angles form a linear pair, we can write

$(7x)^\circ + (11x)^\circ = 180^\circ$

$(18x)^\circ = 180$

Two angles are $7 \times 10 = 70^\circ$ and $11 \times 10 = 110^\circ$

3. Angles $^\angle ABC$ and $^\angle DBC$ form a linear pair of angles. Find the measure of $\angle ABC$ in the following figure.

linear pair of angles with one angle measuring 48º

Since the angles form a linear pair, they are supplementary.

$m\angle ABC + m\angle DBC = 180^\circ$

$m\angle ABC + 48^\circ = 180^\circ$

$m\angle ABC = 180^\circ \;-\;  48^\circ$

$m^\circ ABC = 132^\circ$

4. If one angle in the linear pair is $40^\circ$ , then find the other angle. 

Measure of one angle $= 40^\circ$

Since the angles form a linear pair, they add up to $180^\circ$.

Measure of the other angle $= 180^\circ \;–\; 40^\circ = 140^\circ$

5. One angle of the linear pair is thrice the other. Find the angles.

Let one angle be $x^\circ$ and the other angle be $3x^\circ$. 

Since they form a linear pair, we write

$x^\circ + 3x^\circ = 180^\circ$

$4x^\circ = 180^\circ$

$x^\circ = 45^\circ$

Thus, the angles are $45^\circ$ and $135^\circ$.

6. Do angles that measure $107^\circ$ and $72^\circ$ form a linear pair?

Solution: 

Angles in a linear pair are supplementary. So, if two angles are not supplementary, they are not a linear pair of angles.

$107^\circ + 72^\circ = 179^\circ 180^\circ$

They do not form a linear pair. 

Attend this quiz & Test your knowledge.

Which of the following is the linear pair of $75^\circ$?

What will be the value of x.

Linear Pair of Angles: Definition, Axioms, Examples, Facts, FAQs

Which of the following pairs of angles form a linear pair?

Linear Pair of Angles: Definition, Axioms, Examples, Facts, FAQs

What will be the value of k in the following figure?

Linear Pair of Angles: Definition, Axioms, Examples, Facts, FAQs

How many angles are there in a linear pair?

Only two angles can be found in a linear pair.

Are linear pairs of angles congruent?

Linear pairs of angles are not congruent. When the measure of each of the angles in a linear pair is $90^\circ$, a linear pair of angles are congruent.

Are the linear pair of angles always supplementary?

Supplementary is one of the necessary conditions for angles to be a linear pair. Hence, linear pairs are always supplementary. A linear pair forms a straight angle that measures $180^\circ$.

What is the difference between a linear pair of angles and complementary angles?

A linear pair are two adjacent angles that sum to $180^\circ$. On the other hand, complementary angles are the angles that sum up to $90^\circ$. Complementary angles need not be adjacent.

What is the difference between a linear pair of angles and vertical angles?

A linear pair are two adjacent angles that sum to $180^\circ$. They share a common vertex and a common arm.

Vertical angles are the angles that are opposite angles formed when two lines intersect each other. They only share a common vertex. Vertical angles are always congruent.

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Linear Pair

Related Pages More Lessons for High School Regents Exam Math Worksheets

These lessons cover High School Math based on the topics required for the Regents Exam conducted by NYSED. Here, we look at linear pairs.

What are Linear Pairs? Linear Pairs are two adjacent angles whose non common sides form a straight line. Linear pairs are supplementary angles i.e. they add up to 180°.

The following diagrams show examples of Linear Pairs. Scroll down the page for more examples and solutions on how to identify and use Linear Pairs.

Linear Pairs

Linear Pairs and Vertical Angles This video describes linear pairs and vertical angles. It also solves a few example problems to help you understand these concepts better.

Linear Pair Review Example of how to find the measure of the angles in a linear pair.

Linear Pair Angles An example of solving for a variable for linear pair angles

Solving for Linear Pair and Vertical Angles This video that explains how to solve for the values of x and y when given angles directly across from each other (vertical angles) and next to each other (linear pair angles).

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1.15: Supplementary Angles

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Two angles that add to 180 degrees and when adjacent form a straight line.

Linear Pairs

Two angles are adjacent if they have the same vertex, share a side, and do not overlap. \(\angle PSQ\) and \(\angle QSR\) are adjacent.

f-d_948f9829eba73297972a091b94dd4eb5a471d8e63bb9756af1d4e81f+IMAGE_TINY+IMAGE_TINY.png

A linear pair is two angles that are adjacent and whose non-common sides form a straight line. If two angles are a linear pair, then they are supplementary (add up to \(180^{\circ}\)). \(\angle PSQ\) and \(\angle QSR\) are a linear pair.

f-d_23d540e9be461c6e22261aa41532c5d3b85f23f2ca346d29a61c4357+IMAGE_TINY+IMAGE_TINY.png

What if you were given two angles of unknown size and were told they form a linear pair? How would you determine their angle measures?

For Example \(\PageIndex{1}\) and \(\PageIndex{2}\) , use the diagram below. Note that \(\overline{NK} \perp \overleftrightarrow{IL}\).

f-d_ddbfde705abde0cdc132cdfcd782bec102f8b34c4b23a3ab4f4862ca+IMAGE_TINY+IMAGE_TINY.png

Example \(\PageIndex{1}\)

Name one linear pair of angles.

\(\angle MNL\) and \(\angle LNJ\)

Example \(\PageIndex{2}\)

What is \(m \angle INL\)?

\(180^{\circ}\)

Example \(\PageIndex{3}\)

What is the measure of each angle?

f-d_8bcec55351da12bf7020e7186bc467135e5d4365db11b290a133521d+IMAGE_TINY+IMAGE_TINY.png

These two angles are a linear pair, so they add up to \(180^{\circ}\).

\((7q−46)^{\circ}+(3q+6)^{\circ}=180^{\circ}\)

\(10q−40^{\circ}=220^{\circ}\)

\(10q=180^{\circ}\)

\(q=22^{\circ}\)

Plug in q to get the measure of each angle.

\(m \angle ABD=7(22^{\circ})−46^{\circ}=108^{\circ} \)

\(m \angle DBC=180^{\circ}−108^{\circ}=72^{\circ}\)

Example \(\PageIndex{4}\)

Are \(\angle CDA\) and \(\angle DAB\) a linear pair? Are they supplementary?

f-d_3c72a69d58607c8bcd2ec43af9f9d1233f828398c238e97644cc7e0d+IMAGE_TINY+IMAGE_TINY.png

The two angles are not a linear pair because they do not have the same vertex. They are supplementary because they add up to \(180^{\circ}: 120^{\circ}+60^{\circ}=180^{\circ}\).

Example \(\PageIndex{5}\)

Find the measure of an angle that forms a linear pair with \(\angle MRS\) if \(m \angle MRS\) is \(150^{\circ}\).

Because linear pairs have to add up to \(180^{\circ}\), the other angle must be \(180^{\circ}−150^{\circ}=30^{\circ}\).

For 1-5, determine if the statement is true or false.

  • Linear pairs are congruent.
  • Adjacent angles share a vertex.
  • Adjacent angles overlap.
  • Linear pairs are supplementary.
  • Supplementary angles form linear pairs.

For exercise 6, find the value of \(x\).

f-d_2bc8de1b4f6a1e34f87fba9f38597c9a35ca4a34426964c6968767c7+IMAGE_TINY+IMAGE_TINY.png

Find the measure of an angle that forms a linear pair with \(\angle MRS\) if \(m \angle MRS\) is:

  • \(61^{\circ}\)
  • \(23^{\circ}\)
  • \(114^{\circ}\)
  • \(7^{\circ}\)
  • \(179^{\circ}\)
  • \(z^{\circ}\)

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.9.

Additional Resources

Interactive Element

Video: Complementary, Supplementary, and Vertical Angles

Activities: Supplementary Angles Discussion Questions

Study Aids: Angles Study Guide

Practice: Supplementary Angles

Real World: Supplementary Angles

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Angle Relationships Simply Explained w/ 11+ Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s lesson, you’re going to learn all about angle relationships and their measures.

Jenn (B.S., M.Ed.) of Calcworkshop® introducing angle relationships

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’ll walk through 11 step-by-step examples to ensure mastery.

Let’s dive in!

Angle Pair Relationship Names

In Geometry , there are five fundamental angle pair relationships:

  • Complementary Angles
  • Supplementary Angles
  • Adjacent Angles
  • Linear Pair
  • Vertical Angles

1. Complementary Angles

Complementary angles are two positive angles whose sum is 90 degrees.

For example, complementary angles can be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two acute angles, like ∠MNP and ∠EFG, whose sum is equal to 90 degrees. Both of these graphics represent pairs of complementary angles.

complementary angles example

Complementary Angles Example

2. Supplementary Angles

Supplementary angles are two positive angles whose sum is 180 degrees.

For example, supplementary angles may be adjacent, as seen in with ∠ABD and ∠CBD in the image below. Or they can be two angles, like ∠MNP and ∠KLR, whose sum is equal to 180 degrees. Both of these graphics represent pairs of supplementary angles.

supplementary angles example

Supplementary Angles Example

What is important to note is that both complementary and supplementary angles don’t always have to be adjacent angles.

3. Adjacent Angles

Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.

Angles 1 and 2 are adjacent angles because they share a common side.

adjacent angles examples

Adjacent Angles Examples

And as Math is Fun so nicely points out, a straightforward way to remember Complementary and Supplementary measures is to think:

C is for Corner of a Right Angle (90 degrees) S is for Straight Angle (180 degrees)

Now it’s time to talk about my two favorite angle-pair relationships: Linear Pair and Vertical Angles.

4. Linear Pair

A linear pair is precisely what its name indicates. It is a pair of angles sitting on a line! In fact, a linear pair forms supplementary angles.

Because, we know that the measure of a straight angle is 180 degrees, so a linear pair of angles must also add up to 180 degrees.

∠ABD and ∠CBD form a linear pair and are also supplementary angles, where ∠1 + ∠2 = 180 degrees.

linear pair example

Linear Pair Example

5. Vertical Angles

Vertical angles are two nonadjacent angles formed by two intersecting lines or opposite rays.

Think of the letter X. These two intersecting lines form two sets of vertical angles (opposite angles). And more importantly, these vertical angles are congruent.

In the accompanying graphic, we see two intersecting lines, where ∠1 and ∠3 are vertical angles and are congruent. And ∠2 and ∠4 are vertical angles and are also congruent.

vertical angles examples

Vertical Angles Examples

Together we are going to use our knowledge of Angle Addition, Adjacent Angles, Complementary and Supplementary Angles, as well as Linear Pair and Vertical Angles to find the values of unknown measures.

Angle Relationships – Lesson & Examples (Video)

  • Introduction to Angle Pair Relationships
  • 00:00:15 – Overview of Complementary, Supplementary, Adjacent, and Vertical Angles and Linear Pair
  • Exclusive Content for Member’s Only
  • 00:06:29 – Use the diagram to solve for the unknown angle measures (Examples #1-8)
  • 00:19:05 – Find the measure of each variable involving Linear Pair and Vertical Angles (Examples #9-12)
  • Practice Problems with Step-by-Step Solutions
  • Chapter Tests with Video Solutions

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Linear algebra

Unit 1: vectors and spaces, unit 2: matrix transformations, unit 3: alternate coordinate systems (bases).

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What is a Linear Pair in Geometry – Understanding Angles and Their Relationships

What is a Linear Pair in Geometry Understanding Angles and Their Relationships

In geometry, a linear pair is a concept involving two adjacent angles that share a common arm and whose non-common arms form a straight line .

To be considered a linear pair , these two angles must add up to $180^\circ$ , which means they are supplementary angles .

This is a fundamental concept, as it is closely related to the properties of lines and angles that form the basis for much more complex geometric reasoning. Understanding a linear pair is crucial for grasping how various angles interact within different geometric shapes and designs.

As I explore this topic, I will also discuss how the definition of a linear pair helps us find unknown angles and solve problems involving parallel lines cut by a transversal.

When we see two angles that make a linear pair , we can confidently say their measures sum up to a straight angle , which is essentially a straight line . Let’s dive into the fascinating world of angles and discover why linear pairs are so important in geometry.

What are Linear Pairs in Geometry?

In geometry, I often encounter the concept of linear pairs of angles .

Two adjacent angles forming a straight line, totaling 180 degrees

 These pairs consist of two adjacent angles that have a common vertex and a common arm , while their non-common arms are straight lines on opposite sides. It’s key to know that these angles are also known as supplementary angles because they always add up to a straight angle, which is exactly 180° .

Interestingly, axioms related to linear pairs state that if two angles form a linear pair , then they are supplementary.

This is what’s called the linear pair postulate or sometimes the linear pair axiom . So, whenever I see two angles that share a vertex and a side, and their other sides form a straight line, I can confidently say they form a linear pair.

Here is an illustration of what I mean by linear pairs:

As an example, suppose that $\angle ABC$ and $\angle CBD$ form a linear pair. This means $\angle ABC + \angle CBD = 180°$. The inference I can draw from this is simple: if I know one of the angles, I can effortlessly find the other by subtracting the known angle from 180° .

To sum it up, linear pairs are a foundational concept in understanding how angles interact when lines intersect, and knowing about them gives me a better grasp of geometric principles.

Exploring Linear Pair Examples

When I explore geometry, I often encounter a concept known as the linear pair . Essentially, a linear pair consists of a duo of adjacent angles formed when two lines intersect.

These angles share a common vertex and a common ray , while their non- adjacent line segments span out to form a straight line .

For instance, let’s consider two rays , $\overrightarrow{BA}$ and $\overrightarrow{BC}$, emanating from a single point ( B ) and stretching infinitely in opposite directions.

Now, if a third ray $ \overrightarrow{BD}$ bisects the straight angle created by $ \overrightarrow{BA} $ and $\overrightarrow{BC}$, the angles $\angle ABD $ and $\angle DBC$ illustrate a linear pair .

Table: Linear Pair Properties

It’s crucial to practice identifying and understanding linear pairs . They serve as fundamental components in solving various geometric problems, especially when dealing with types of angles or proving that certain angles are complementary or the vertical angles .

One way to solve for an unknown angle in a linear pair is by using the supplementary property, since if one angle measures ( x ) degrees, the other is ( 180^{\circ} – x ).

Moreover, a solid grasp of linear pairs facilitates recognizing patterns within geometric figures such as triangles and line segments , opening up a realm of practice problems to reinforce my understanding.

When I approach these examples, I not only strengthen my ability to solve for unknowns but also begin to see the interconnectedness of geometric principles, like how linear pairs relate to opposite rays and the broader topic of types of angles .

In geometry , the concept of a linear pair of angles is fundamental and contributes to my understanding of angular relationships.

A linear pair consists of two adjacent angles with their non-adjacent sides forming a straight line, which is mathematically expressed as $\overleftrightarrow{AD}$ in the given diagram.

This characteristic alignment stipulates that the angles are supplementary, meaning the sum of their measures is equal to $\boldsymbol{180^\circ}$ , or $\angle ABC + \angle DBC = 180^\circ$ .

The supremacy of the linear pair axiom is underscored by its wide applicability. For instance, it assists me in deducing unknown angle measures when provided with at least one angle of the pair.

Remembering that each angle is poised to complement the other up to $\boldsymbol{180^\circ}$ is handy in solving various geometric problems.

It’s also crucial for me to properly grasp that the converse of the linear pair axiom holds; when two angles are supplementary, it typically suggests they form a linear pair as long as they are adjacent and share a common vertex and side.

By appreciating these intricacies of linear pairs , my proficiency in geometry is significantly enhanced, and I’m equipped to tackle related problems with confidence.

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Linear Pair Of Angles

Linear pair of angles are formed when two lines intersect each other at a single point. The angles are said to be linear if they are adjacent to each other after the intersection of the two lines. The sum of angles of a linear pair is always equal to 180°. Such angles are also known as supplementary angles. The adjacent angles are the angles which have a common vertex. Hence, here as well the linear angles have a common vertex. Also, there will be a common arm which represents both the angles. A real-life example of a linear pair is a ladder which is placed against a wall, forms linear angles at the ground.

Linearity represents one which is straight. So here also, linear angles are the one which is formed into a straight line. The pair of adjacent angles here are constructed on a line segment,  but not all adjacent angles are linear. Hence, we can also say, that linear pair of angles is the adjacent angles whose non-common arms are basically opposite rays.

Explanation for Linear Pair of Angles

When the angle between the two lines is 180°, they form a straight angle . A straight angle is just another way to represent a straight line. A straight line can be visualized as a circle with an infinite radius. A line segment is any portion of a line which has two endpoints. Also, a portion of any line with only one endpoint is called a ray.

The figure shown below represents a line segment AB and the two arrows at the end indicate a line.

Line segment

If a point O is taken anywhere on the line segment AB as shown, then the angle between the two line segments AO and OB is a straight angle i.e. 180°.

Straight angle

The angles which are formed at O are ∠ POB and ∠ POA . It is known that the angle between the two line segments AO and OB is 180°. therefore, the angles ∠ POB and ∠ POA add up to 180°.

Thus, ∠ POB + ∠ POA = ∠ AOB = 180°

∠POB and ∠POA are adjacent to each other and when the sum of adjacent angles is 180° then such angles form linear pair of angles.

The above discussion can be stated as an axiom.

Also, read:

Axiom 1: If a ray stands on a line then the adjacent angles form a linear pair of angles.

Linear Pair of angles axiom

In the figure above, all the line segments pass through the point O as shown. As the ray OA  lies on the line segment CD, angles ∠ AOD and ∠ AOC form a linear pair. Similarly, ∠ QOD and ∠ POD form a linear pair and so on.

The converse of the stated axiom is also true, which can also be stated as the following axiom.

Axiom 2: If two angles form a linear pair, then uncommon arms of both the angles form a straight line.

Linear Pair Angles Axiom 2

In the figure shown above, only the last one represents a linear pair, as the sum of the adjacent angles is 180°. Therefore, AB represents a line. The other two pairs of angles are adjacent but they do not form a linear pair. They do not form a straight line.

The two axioms mentioned above form the Linear Pair Axioms and are very helpful in solving various mathematical problems.

Video Lesson on Types of Angles

linear pair algebra 1 definition

Suppose two angles ∠AOC and ∠ BOC form a linear pair at point O in a line segment AB. If the difference between the two angles is 60°. Then find both the angles.

Solution: Given, ∠AOC and ∠ BOC form a linear pair

So, ∠AOC + ∠ BOC =180° ………(1)

Also given,

∠AOC – ∠ BOC = 60° ………(2)

Adding eq. 1 and 2, we get;

2∠AOC = 180° + 60° = 240°

∠AOC = 240°/2 = 120°

Now putting the value of ∠AOC in equation 1, we get;

∠BOC = 180° – ∠AOC = 180° – 120°

There is a lot more to learn about lines and angles . To know more about properties of pair of angles, download BYJU’S – The Learning App from Google Play Store.

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Parallel and perpendicular lines

  • Choose the parallel line
  • Perpendicular line

If two non-vertical lines that are in the same plane has the same slope, then they are said to be parallel. Two parallel lines won't ever intersect.

picture31

If two non-vertical lines in the same plane intersect at a right angle then they are said to be perpendicular. Horizontal and vertical lines are perpendicular to each other i.e. the axes of the coordinate plane.

Compare the slope of the perpendicular lines

linear pair algebra 1 definition

The slope of the red line:

$$m_{1}=\frac{-3-1}{2-( -2  )}=\frac{-4}{4}=-1$$

The slope of the blue line

$$m_{2}=\frac{2-\left ( -2 \right )}{3-\left ( -1 \right )}=\frac{4}{4}=1$$

The slopes of two perpendicular lines are negative reciprocals.

The product of the slopes of two perpendicular lines is -1 since

$$m\cdot -\frac{1}{m}=-1,\: \: where\: \: m_{1}=m\: \: and\: \: m_{2}=-\frac{1}{m}$$

Video lesson

Are these two line parallel?

picture33

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Linear Equations

A linear equation is an equation for a straight line

These are all linear equations:

Let us look more closely at one example:

Example: y = 2x + 1 is a linear equation:

The graph of y = 2x+1 is a straight line

  • When x increases, y increases twice as fast , so we need 2x
  • When x is 0, y is already 1. So +1 is also needed
  • And so: y = 2x + 1

Here are some example values:

Check for yourself that those points are part of the line above!

Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

Examples: These are linear equations:

But the variables (like "x" or "y") in Linear Equations do NOT have:

  • Exponents (like the 2 in x 2 )
  • Square roots , cube roots , etc

Examples: These are NOT linear equations:

Slope-intercept form.

The most common form is the slope-intercept equation of a straight line :

Example: y = 2x + 1

  • Slope: m = 2
  • Intercept: b = 1

Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

Example: y − 3 = (¼)(x − 2)

It is in the form y − y 1 = m(x − x 1 ) where:

General Form

And there is also the General Form of the equation of a straight line:

Example: 3x + 2y − 4 = 0

It is in the form Ax + By + C = 0 where:

There are other, less common forms as well.

As a Function

Sometimes a linear equation is written as a function , with f(x) instead of y :

And functions are not always written using f(x):

The Identity Function

There is a special linear function called the "Identity Function":

And here is its graph:

It is called "Identity" because what comes out is identical to what goes in:

Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

No matter what value of "x", f(x) is always equal to some constant value.

Using Linear Equations

You may like to read some of the things you can do with lines:

  • Finding the Midpoint of a Line Segment
  • Finding Parallel and Perpendicular Lines
  • Finding the Equation of a Line from 2 Points

Linear Algebra

Linear algebra is a branch of mathematics that deals with linear equations and their representations in the vector space using matrices. In other words, linear algebra is the study of linear functions and vectors. It is one of the most central topics of mathematics. Most modern geometrical concepts are based on linear algebra.

Linear algebra facilitates the modeling of many natural phenomena and hence, is an integral part of engineering and physics. Linear equations, matrices, and vector spaces are the most important components of this subject. In this article, we will learn more about linear algebra and the various associated topics.

What is Linear Algebra?

Linear algebra can be defined as a branch of mathematics that deals with the study of linear functions in vector spaces. When information related to linear functions is presented in an organized form then it results in a matrix. Thus, linear algebra is concerned with vector spaces, vectors, linear functions, the system of linear equations, and matrices. These concepts are a prerequisite for sister topics such as geometry and functional analysis.

Linear Algebra Definition

The branch of mathematics that deals with vectors, matrics, finite or infinite dimensions as well as a linear mapping between such spaces is defined as linear algebra. It is used in both pure and applied mathematics along with different technical forms such as physics, engineering, natural sciences, etc.

Branches of Linear Algebra

Linear algebra can be categorized into three branches depending upon the level of difficulty and the kind of topics that are encompassed within each. These are elementary, advanced, and applied linear algebra. Each branch covers different aspects of matrices, vectors, and linear functions.

Elementary Linear Algebra

Elementary linear algebra introduces students to the basics of linear algebra. This includes simple matrix operations, various computations that can be done on a system of linear equations, and certain aspects of vectors. Some important terms associated with elementary linear algebra are given below:

Scalars - A scalar is a quantity that only has magnitude and not direction. It is an element that is used to define a vector space. In linear algebra, scalars are usually real numbers.

Vectors - A vector is an element in a vector space. It is a quantity that can describe both the direction and magnitude of an element.

Vector Space - The vector space consists of vectors that may be added together and multiplied by scalars.

Matrix - A matrix is a rectangular array wherein the information is organized in the form of rows and columns. Most linear algebra properties can be expressed in terms of a matrix.

Matrix Operations - These are simple arithmetic operations such as addition , subtraction , and multiplication that can be conducted on matrices.

Advanced Linear Algebra

Once the basics of linear algebra have been introduced to students the focus shifts on more advanced concepts related to linear equations, vectors, and matrices. Certain important terms that are used in advanced linear algebra are as follows:

Linear Transformations - The transformation of a function from one vector space to another by preserving the linear structure of each vector space.

Inverse of a Matrix - When an inverse of a matrix is multiplied with the given original matrix then the resultant will be the identity matrix. Thus, A -1 A = I.

Eigenvector - An eigenvector is a non-zero vector that changes by a scalar factor (eigenvalue) when a linear transformation is applied to it.

Linear Map - It is a type of mapping that preserves vector addition and vector multiplication.

Applied Linear Algebra

Applied linear algebra is usually introduced to students at a graduate level in fields of applied mathematics, engineering, and physics. This branch of algebra is driven towards integrating the concepts of elementary and advanced linear algebra with their practical implications. Topics such as the norm of a vector, QR factorization, Schur's complement of a matrix, etc., fall under this branch of linear algebra.

Linear Algebra Topics

The topics that come under linear algebra can be classified into three broad categories. These are linear equations, matrices, and vectors. All these three categories are interlinked and need to be understood well in order to master linear algebra. The topics that fall under each category are given below.

Linear Equations

A linear equation is an equation that has the standard form \(a_{1}x_{1} + a_{2}x_{2} + ... + a_{n}x_{n}\). It is the fundamental component of linear algebra. The topics covered under linear equations are as follows:

  • Linear Equations in One variable
  • Linear Equations in Two Variables
  • Simultaneous Linear Equations
  • Solving Linear Equations
  • Solutions of a Linear Equation
  • Graphing Linear Equations
  • Applications of Linear equations
  • Straight Line

In linear algebra, there can be several operations that can be performed on vectors such as multiplication , addition, etc. Vectors can be used to describe quantities such as the velocity of moving objects. Some crucial topics encompassed under vectors are as follows:

  • Types of Vectors
  • Dot Product
  • Cross Product
  • Addition of Vectors

A matrix is used to organize data in the form of a rectangular array. It can be represented as \(A_{m\times n}\). Here, m represents the number of rows and n denotes the number of columns in the matrix. In linear algebra, a matrix can be used to express linear equations in a more compact manner. The topics that are covered under the scope of matrices are as follows:

  • Matrix Operations
  • Determinant
  • Transpose of a Matrix
  • Types of a Matrix

Linear Algebra Formula

Formulas form an important part of linear algebra as they help to simplify computations. The key to solving any problem in linear algebra is to understand the formulas and associated concepts rather than memorize them. The important linear algebra formulas can be broken down into 3 categories, namely, linear equations, vectors, and matrices.

Linear Equations: The important linear equation formulas are listed as follows:

  • General form: ax + by = c
  • Slope Intercept Form : y = mx + b
  • a + b = b + a
  • a + 0 = 0 + a = a

Vectors: If there are two vectors \(\overrightarrow{u}\) = (\(u_{1}\), \(u_{2}\), \(u_{3}\)) and \(\overrightarrow{v}\) = (\(v_{1}\), \(v_{2}\), \(v_{3}\)) then the important vector formulas associated with linear algebra are given below.

  • \(\overrightarrow{u} + \overrightarrow{v} = (u_{1}+v_{1}, u_{2}+v_{2}, u_{3}+v_{3})\)
  • \(\overrightarrow{u} - \overrightarrow{v} = (u_{1}-v_{1}, u_{2}-v_{2}, u_{3}-v_{3})\)
  • \(\left \| u \right \| = \sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}}\)
  • \(\overrightarrow{u}.\overrightarrow{v} = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}\)
  • \(\overrightarrow{u}\times \overrightarrow{v} = (u_{2}v_{3}-u_{3}v_{2}, u_{3}v_{1}-u_{1}v_{3}, u_{1}v_{2}-u_{2}v_{1})\)

Matrix: If there are two square matrices given by A and B where the elements are \(a_{ij}\) and \(b_{ij}\) respectively, then the following important formulas are used in linear algebra:

Linear Algebra

  • C = A + B, where \(c_{ij}\) = \(a_{ij}\) + \(b_{ij}\)
  • C = A - B, where \(c_{ij}\) = \(a_{ij}\) - \(b_{ij}\)
  • kA = k\(a_{ij}\)
  • C = AB = \(\sum_{k = 1}^{n}a_{ik}b_{kj}\)

Linear Algebra and its Applications

Linear algebra is used in almost every field. Simple algorithms also make use of linear algebra topics such as matrices. Some of the applications of linear algebra are given as follows:

  • Signal Processing - Linear algebra is used in encoding and manipulating signals such as audio and video signals. Furthermore, it is required in the analysis of such signals.
  • Linear Programming - It is an optimizing technique that is used to determine the best outcome of a linear function.
  • Computer Science - Data scientists use several linear algebra algorithms to solve complicated problems.
  • Prediction Algorithms - Prediction algorithms use linear models that are developed using concepts of linear algebra.

Related Articles:

  • Introduction to Graphing
  • One Variable Linear Equations and Inequalities
  • Resolving a Vector into Components

Important Notes on Linear Algebra

  • Linear algebra is concerned with the study of three broad subtopics - linear functions, vectors, and matrices
  • Linear algebra can be classified into 3 categories. These are elementary, advanced, and applied linear algebra.
  • Elementary linear algebra is concerned with the introduction to linear algebra. Advanced linear algebra builds on these concepts. Applied linear algebra applies these concepts to real-life situations.

Linear Algebra Examples

  • Example 1: Using linear algebra add these two matrices. A = \(\begin{bmatrix} 5 & 6\\ 2& 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 7\\ 5& 4 \end{bmatrix}\) Solution: C = A + B C = \(\begin{bmatrix} 5 & 6\\ 2& 1 \end{bmatrix}\) + \(\begin{bmatrix} 3 & 7\\ 5& 4 \end{bmatrix}\) C = \(\begin{bmatrix} 8 & 13\\ 7& 5 \end{bmatrix}\) Answer: C = \(\begin{bmatrix} 8 & 13\\ 7& 5 \end{bmatrix}\)
  • Example 2: Subtract the two vectors \(\vec{u}\) = (3, 7, 1) and \(\vec{v}\) = (6, 2, 8) using linear algebra Solution: \(\vec{u}\) - \(\vec{v}\) = (-3, 5, -7) Answer: (-3, 5, -7)
  • Example 3: Solve the equations: x + 3 = 2(y - 1) and y + 1 = 5x Solution: Solving by substitution, x + 3 = 2(y - 1) x = 2y - 5 Putting this value in the second equation, y + 1 = 5 (2y - 5) y = 26 / 9 Now y + 1 = 5x (26 / 9) + 1 = 5x x = 7 / 9 Answer: x = 7 / 9, y = 26 / 9

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Practice Questions on Linear Algebra

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FAQs on Linear Algebra

What is the meaning of linear algebra.

Linear algebra is a branch of mathematics that deals with the study of linear functions , vectors, matrices, and other associated aspects.

Is Linear Algebra Difficult?

Linear algebra is a very vast branch of mathematics. However, with regular practice and instilling a strong conceptual foundation solving questions will be very easy.

What are the Prerequisites for Linear Algebra?

It is necessary to have a strong foundation regarding the properties of numbers and how to perform calculations before starting linear algebra.

What is a Subspace in Linear Algebra?

A vector space that is entirely contained in another vector space is known as a subspace in linear algebra.

How to Study Linear Algebra?

The first step is to instill a strong foundation in elementary algebra. Understanding concepts and regular revision of formulas are also crucial before moving on to advanced algebra. It is equally necessary to solve practice questions of various levels to succeed in this subject.

Is Linear Algebra Harder than Calculus?

Linear algebra serves as a prerequisite for calculus . It is important to develop deep-seated knowledge of this subject before moving on to calculus. Both subjects are easy as long as concepts are clear and sums are practiced regularly.

What is Linear Algebra Used for?

Linear algebra is used in several industries such as computer science, engineering as well as physics to create linear models using the algorithms outlined in this subject.

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Mathematics LibreTexts

5.1: Linear Transformations

  • Last updated
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  • Page ID 14524

  • Ken Kuttler
  • Brigham Young University via Lyryx
  • Understand the definition of a linear transformation, and that all linear transformations are determined by matrix multiplication.

Recall that when we multiply an \(m\times n\) matrix by an \(n\times 1\) column vector, the result is an \(m\times 1\) column vector. In this section we will discuss how, through matrix multiplication, an \(m \times n\) matrix transforms an \(n\times 1\) column vector into an \(m \times 1\) column vector.

Recall that the \(n \times 1\) vector given by \[\vec{x} = \left [ \begin{array}{r} x_1 \\ x_2\\ \vdots \\ x_n \end{array} \right ]\nonumber \] is said to belong to \(\mathbb{R}^n\), which is the set of all \(n \times 1\) vectors. In this section, we will discuss transformations of vectors in \(\mathbb{R}^n.\)

Consider the following example.

Example \(\PageIndex{1}\): A Function Which Transforms Vectors

Consider the matrix \(A = \left [ \begin{array}{ccc} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] .\) Show that by matrix multiplication \(A\) transforms vectors in \(\mathbb{R}^3\) into vectors in \(\mathbb{R}^2\).

First, recall that vectors in \(\mathbb{R}^3\) are vectors of size \(3 \times 1\), while vectors in \(\mathbb{R}^{2}\) are of size \(2 \times 1\). If we multiply \(A\), which is a \(2 \times 3\) matrix, by a \(3 \times 1\) vector, the result will be a \(2 \times 1\) vector. This what we mean when we say that \(A\) transforms vectors.

Now, for \(\left [ \begin{array}{c} x \\ y \\ z \end{array} \right ]\) in \(\mathbb{R}^3\), multiply on the left by the given matrix to obtain the new vector. This product looks like \[\left [ \begin{array}{rrr} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] \left [ \begin{array}{r} x \\ y \\ z \end{array} \right ] = \left [ \begin{array}{c} x+2y \\ 2x+y \end{array} \right ]\nonumber \] The resulting product is a \(2 \times 1\) vector which is determined by the choice of \(x\) and \(y\). Here are some numerical examples. \[\left [ \begin{array}{ccc} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] \left [ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right ] = \ \left [ \begin{array}{c} 5 \\ 4 \end{array} \right ]\nonumber \] Here, the vector \(\left [ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right ]\) in \(\mathbb{R}^3\) was transformed by the matrix into the vector \(\left [ \begin{array}{c} 5 \\ 4 \end{array}\right ]\) in \(\mathbb{R}^2\).

Here is another example: \[\left [ \begin{array}{rrr} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array} \right ] \left [ \begin{array}{r} 10 \\ 5 \\ -3 \end{array} \right ] = \ \left [ \begin{array}{r} 20 \\ 25 \end{array} \right ]\nonumber \]

The idea is to define a function which takes vectors in \(\mathbb{R}^{3}\) and delivers new vectors in \(\mathbb{R}^{2}.\) In this case, that function is multiplication by the matrix \(A\).

Let \(T\) denote such a function. The notation \(T:\mathbb{R}^{n}\mapsto \mathbb{R}^{m}\) means that the function \(T\) transforms vectors in \(\mathbb{R}^{n}\) into vectors in \(\mathbb{R}^{m}\). The notation \(T(\vec{x})\) means the transformation \(T\) applied to the vector \(\vec{x}\). The above example demonstrated a transformation achieved by matrix multiplication. In this case, we often write \[T_{A}\left( \vec{x}\right) =A \vec{x}\nonumber \] Therefore, \(T_{A}\) is the transformation determined by the matrix \(A\). In this case we say that \(T\) is a matrix transformation.

Recall the property of matrix multiplication that states that for \(k\) and \(p\) scalars, \[A\left( kB+pC\right) =kAB+pAC\nonumber \] In particular, for \(A\) an \(m\times n\) matrix and \(B\) and \(C,\) \(n\times 1\) vectors in \(\mathbb{R}^{n}\), this formula holds.

In other words, this means that matrix multiplication gives an example of a linear transformation, which we will now define.

Definition \(\PageIndex{1}\): Linear Transformation

Let \(T:\mathbb{R}^{n}\mapsto \mathbb{R}^{m}\) be a function, where for each \(\vec{x} \in \mathbb{R}^{n},T\left(\vec{x}\right)\in \mathbb{R}^{m}.\) Then \(T\) is a linear transformation if whenever \(k ,p\) are scalars and \(\vec{x}_1\) and \(\vec{x}_2\) are vectors in \(\mathbb{R}^{n}\) \(( n\times 1\) vectors\(),\) \[T\left( k \vec{x}_1 + p \vec{x}_2 \right) = kT\left(\vec{x}_1\right)+ pT\left(\vec{x}_{2} \right)\nonumber \]

Example \(\PageIndex{2}\): Linear Transformation

Let \(T\) be a transformation defined by \(T:\mathbb{R}^3\to\mathbb{R}^2\) is defined by \[T\left [\begin{array}{c} x \\ y \\ z \end{array}\right ] = \left [\begin{array}{c} x+y \\ x-z \end{array}\right ] \mbox{ for all } \left [\begin{array}{c} x \\ y \\ z \end{array}\right ] \in\mathbb{R}^3\nonumber \] Show that \(T\) is a linear transformation.

By Definition \(\PageIndex{1}\) we need to show that \(T\left( k \vec{x}_1 + p \vec{x}_2 \right) = kT\left(\vec{x}_1\right)+ pT\left(\vec{x}_{2} \right)\) for all scalars \(k,p\) and vectors \(\vec{x}_1, \vec{x}_2\). Let \[\vec{x}_1 = \left [\begin{array}{c} x_1 \\ y_1 \\ z_1 \end{array}\right ], \vec{x}_2 = \left [\begin{array}{c} x_2 \\ y_2 \\ z_2 \end{array}\right ]\nonumber \] Then \[\begin{aligned} T\left( k \vec{x}_1 + p \vec{x}_2 \right) &= T \left( k \left [\begin{array}{c} x_1 \\ y_1 \\ z_1 \end{array}\right ] + p \left [\begin{array}{c} x_2 \\ y_2 \\ z_2 \end{array}\right ] \right) \\ &= T \left( \left [\begin{array}{c} kx_1 \\ ky_1 \\ kz_1 \end{array}\right ] + \left [\begin{array}{c} px_2 \\ py_2 \\ pz_2 \end{array}\right ] \right) \\ &= T \left( \left [\begin{array}{c} kx_1 + px_2 \\ ky_1 + py_2 \\ kz_1 + pz_2 \end{array}\right ] \right) \\ &= \left [\begin{array}{c} (kx_1 + px_2) + (ky_1 + py_2) \\ (kx_1 + px_2)- (kz_1 + pz_2) \end{array}\right ] \\ &= \left [\begin{array}{c} (kx_1 + ky_1) + (px_2 + py_2) \\ (kx_1 - kz_1) + (px_2 - pz_2) \end{array}\right ] \\ &= \left [\begin{array}{c} kx_1 + ky_1 \\ kx_1 - kz_1 \end{array}\right ] + \left [ \begin{array}{c} px_2 + py_2 \\ px_2 - pz_2 \end{array}\right ] \\ &= k \left [\begin{array}{c} x_1 + y_1 \\ x_1 - z_1 \end{array}\right ] + p \left [ \begin{array}{c} x_2 + y_2 \\ x_2 - z_2 \end{array}\right ] \\ &= k T(\vec{x}_1) + p T(\vec{x}_2) \end{aligned}\nonumber \] Therefore \(T\) is a linear transformation.

Two important examples of linear transformations are the zero transformation and identity transformation. The zero transformation defined by \(T\left( \vec{x} \right) = \vec(0)\) for all \(\vec{x}\) is an example of a linear transformation. Similarly the identity transformation defined by \(T\left( \vec{x} \right) = \vec(x)\) is also linear. Take the time to prove these using the method demonstrated in Example \(\PageIndex{2}\) .

We began this section by discussing matrix transformations, where multiplication by a matrix transforms vectors. These matrix transformations are in fact linear transformations.

Theorem \(\PageIndex{1}\): Matrix Transformations are Linear Transformations

Let \(T:\mathbb{R}^{n}\mapsto \mathbb{R}^{m}\) be a transformation defined by \(T(\vec{x}) = A\vec{x}\). Then \(T\) is a linear transformation.

It turns out that every linear transformation can be expressed as a matrix transformation, and thus linear transformations are exactly the same as matrix transformations.

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