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01. Introduction and Overview

Dated: 10-03-2025

History

Algebra is named after Mohammed Ibn-e-Musa al-Khowarizmi. Around 825, he wrote a book entitled al-jabr u'l muqubalah (the science of reduction and cancellation).
Algebra is branch of Mathematics which deals with the relationships of unknown and known quantities.

Terminology

An algebraic term is a product of a number with one or more variables.

Example

\(4x\) is an algebraic term where \(4\) is called the coefficient and \(x\) is called the variable.

Algebraic Expressions

An expression is a collection of

  • Numbers
  • Variables
  • Positive or negative signs of operations

which make up the mathematical and logical behavior.

Example

\[8x^2 + 9x - 1\]

What is Linear Algebra

It provides tools for analyzing

  • Differential Equations
  • Statistical Processes
  • Physical Phenomena

It creates a formal link between matrix calculus and use of linear or quadratic transformations.

Applications of Linear Algebra

It makes possible to work with large arrays of data and to make sense out of it in a very compact way.
It has applications in different fields such as

  • Computer Graphics
  • Electronics
  • Chemistry
  • Biology
  • Differential Equations
  • Economics
  • Business
  • Psychology
  • Engineering
  • Analytic Geometry
  • Chaos Theory
  • Cryptography
  • Fractal Geometry
  • Game Theory
  • Graph Theory
  • Linear Programming
  • Operations Research

Why Use Linear Algebra

Since linearity is fundamental to any mathematical analysis, this subject lays the foundation for many branches of mathematics.
From experiments, we get a lot of discrete results. Linear algebra provides us tools to deal with these effectively and is used in areas like

  • Physics
  • Fluid Dynamics
  • Signal Processing
  • Numerical Analysis

Objects in Linear Algebra

Vector spaces and their transformations are useful as they cover a broad range of applications.

  1. The solutions of homogeneous systems of linear equations form paradigm examples of vector spaces.
  2. The vectors1 of physics, such as force, as the language suggests, also provide paradigmatic examples.
  3. Solutions to specific systems of differential equations2 also form vector spaces.
  4. Statistics uses linear algebra a lot.
  5. Signal processing uses linear algebra a lot.
  6. Vector spaces appear in number theory including study of field extensions.
  7. Linear algebra motivates abstract algebra.
  8. Vector spaces appear in the study of differential geometry through the tangent bundle of a manifold.
  9. Many mathematical models, especially discrete ones, use matrices to represent critical relationships and processes. These are used in
    • Engineering
    • Economics
    • Social sciences

There are 2 principle aspects to linear algebra

  • Theoretical
  • Computational

The art is to move back and forth from one to another.

References

Read more about notations and symbols.

  1. Read more about vectors. ↩

  2. Read more about differential equations. ↩