From Words to Equations: The Intersection of Sentences and Linear Algebra
Linking Sentences with Linear Algebra Concepts
Linear algebra often begins with the study of systems of linear equations. However, for understanding these systems requires more than just solving equation as it demands familiarity with the language of mathematics. Equations are much like sentences as they convey information. When grouped together as systems, they interact to produce a clearer picture of the world. Let’s dive into this connection using a relatable example: systems of sentences.
Systems of Sentences
Imagine you're solving a puzzle to figure out the colors of your pets. You have one dog and one cat and each is a certain color. The sentences you have form a system and our goal is to gather as much information as possible about your pets.
System 1:
Sentence 1: The cat is grey.
Sentence 2: The dog is orange.
Here, This system is complete as it contains two sentences that provide two distinct pieces of information. It’s as informative as possible.
System 2:
Sentence 1: The dog is orange.
Sentence 2: The dog is orange.
Here, the sentences is repeating themselves. Although the system has two sentences, it is only conveying one piece of information which makes the system redundant.
System 3:
Sentence 1: The dog is orange.
Sentence 2: The dog is white.
Here, This system is contradicting itself as the dog can’t be both orange and white.
Singular vs. Non-Singular Systems
The utility of a system depends on its informativeness as our a system is to convey as much information as possible with these simple sentences
Non-Singular System: A system that conveys as much information as it contains sentences. It’s complete and free of redundancies or contradictions.
Singular System: A system that is either redundant or contradictory and less informative.
By treating equations like sentences, you can approach systems methodically . When systems grow in size, the same principles apply. For example, suppose you have three animals .i.e. a dog, a cat, and a bird and their colors to determine.
System 1:
Sentence 1: The cat is grey.
Sentence 2: The dog is orange.
Sentence 3: The bird is cyan.
Statement:
This system is non-singular as it contains three distinct pieces of information.
System 2:
Sentence 1: The dog is white.
Sentence 2: The dog is white.
Sentence 3: The bird is cyan.
Statement:
This system is singular and redundant as sentences repeat which is reducing the system’s informativeness.
System 3:
Sentence 1: The cat is grey.
Sentence 2: The cat is grey.
Sentence 3: The cat is grey.
Statement:
This system is extremely redundant as all sentences say the same thing.
System 4:
Sentence 1: The dog is black.
Sentence 2: The dog is white.
Sentence 3: The bird is cyan.
Statement:
This system is singular and contradictory as the dog can’t be both orange and white.
Let’s recap the three types of systems we’ve learned
1. Complete and Non-Singular:
A system of linear equations is considered complete and non-singular if each equation adds unique and independent information to the system. This ensures that the equations do not overlap in their contributions and collectively define a single and unique solution. Thus, the system has exactly as many independent equations as there are variables.
2. Redundant and Singular:
A system of linear equations is redundant and singular when some equations are linear combinations of others which means they provide the same information. As a result, the system lacks sufficient independent equations to uniquely determine the solution. So, The system has infinitely many solutions because the equations describe overlapping or coinciding geometrical objects .i.e. lines, planes, etc.
3. Contradictory and Singular:
A system of linear equations is contradictory and singular if the equations conflict with each other and it makes impossible to satisfy all equations simultaneously. In this case, the system has no solution. The system is inconsistent as at least one equation contradicts the others. Geometrically, the equations represent same objects (lines, planes, etc.) that do not intersect.
At a Glance: Three Types of Systems :
| Type of System | Number of Solutions | Reason |
| Complete and Non-Singular | One unique solution | Each equation adds independent information and forms a consistent system. |
| Redundant and Singular | Infinitely many solutions | Equations overlap in information as it is describing the same geometrical object. |
| Contradictory and Singular | No solution | Equations are conflicting each other as leading to inconsistency. |
Why This Matters in Linear Algebra?
Just as systems of sentences convey information about the world , the systems of equations reveal the relationships between variables. The concepts of completeness, redundancy, and contradiction in systems of equations align with the ideas of linear independence, dependence, and inconsistency.
So, understanding these foundational ideas not only helps you solve equations but also prepares you to tackle more advanced problems in linear algebra and its applications. We’ll dive deeper into these concepts in our next blog!
Next time you come across a system of equations, imagine them as pieces of a conversation. Each one sharing a bit of insight. When you put them together, they tell a story and helps you to uncover the full picture and make sense of the relationships they describe.