Understanding Systems of Linear Equations | Real-World Applications and Step-by-Step Examples
In this introductory linear algebra video, you'll learn what a system of linear equations is, why it matters, and how it’s used to solve real-world problems. We begin by explaining the general form of a linear system and then walk through two application-based examples:
• Chemistry Application: Solve a saline solution mixing problem using a system of equations.
• Traffic Flow Application: Analyze how traffic patterns can be modeled with linear systems.

This lesson also previews the problem-solving techniques covered throughout the series, including Gauss-Jordan elimination—a systematic method for solving systems of equations. Whether you're a college student preparing for an exam or someone looking to understand how linear algebra is used in STEM fields, this video provides a clear foundation.
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Solving Systems of Linear Equations – Part One | Types of Solutions and Matrix Representation
This foundational video is Part One in our Linear Algebra series on systems of linear equations. You'll gain a clear understanding of what linear systems are, how to recognize the different types of solutions they can have, and how to represent them using matrices.

Key Topics Covered:
• Basic Concepts: Understand the structure of linear systems and what defines them
• Types of Solutions: Learn how to identify when a system has no solution, one unique solution, or infinitely many solutions—explained through geometric interpretations
• Matrix Representation: Learn how to express a linear system using the coefficient matrix and augmented matrix

This lesson lays the foundation for solving linear systems and prepares you for the next video, where we’ll introduce row operations and examine how to determine system consistency.
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Solving Systems of Linear Equations – Part Two | Row Operations and System Consistency
In this video from our Linear Algebra series, we focus on solving systems of linear equations using elementary row operations and augmented matrices. You’ll learn how to systematically reduce a system to row-echelon form and determine whether a system is consistent or inconsistent, especially when parameters are involved.

What You'll Learn:
• How to solve linear systems using row operations
• How to use augmented matrices to simplify systems
• How to determine if a system has no solution, one solution, or infinitely many solutions
• A complete example demonstrating these methods in action

This lesson builds on the fundamentals of linear systems and prepares you to master techniques like Gaussian and Gauss-Jordan elimination for more complex problems in linear algebra.
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Row Reduction and Echelon Forms | Gauss-Jordan Elimination and Applications
In this lesson from our Linear Algebra series, we explore row reduction techniques and the structure of echelon and reduced echelon forms. You’ll learn how to identify pivot positions and pivot columns, and how to apply the Gauss-Jordan elimination algorithm to solve linear systems.

We work through several example problems to:
• Convert matrices into echelon and reduced echelon forms
• Determine how many solutions a linear system has—none, one, or infinitely many
• Analyze systems with parameters hhh and kkk
• Solve linear systems using Gaussian elimination

To conclude, we apply Gauss-Jordan elimination to find a quadratic interpolating polynomial given three data points—demonstrating how linear algebra techniques extend beyond solving equations.

This video is essential for mastering the mechanics of linear systems and seeing their practical applications.
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Introduction to Vectors in Euclidean Space | Components, Operations, and Applications
This video introduces the fundamentals of vectors in Euclidean space, a key concept in linear algebra. We begin with vectors in R^2 and explain how to write vector components, perform scalar multiplication, and carry out vector addition and subtraction, both algebraically and graphically.

You’ll also learn:
• How these concepts extend to vectors in R^n
• The algebraic properties of vectors across dimensions
• How to evaluate linear combinations of vectors through step-by-step examples
• A real-world application: calculating the center of mass of a thin rectangular plate using vector operations

Whether you're just starting out or reinforcing your understanding, this video lays the foundation for mastering vector operations in linear algebra.
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Linear Independence and Dependence of Vectors | Visual and Algebraic Methods
In this video, you'll learn the core concepts behind linear independence and dependence—essential topics in linear algebra. We start by showing how to visually determine whether two vectors are linearly independent or dependent using geometric intuition.

You'll also explore:
• A step-by-step algebraic approach to test if a set of vectors is linearly independent
• How to set up and interpret vector equations
• Example problems to reinforce each concept
• Four special cases where determining linear dependence or independence becomes straightforward

This video equips you with both visual and computational tools to recognize and evaluate the independence of vectors in R^n, laying the groundwork for deeper study in vector spaces and spanning sets.
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Understanding the Span of a Set of Vectors | Algebraic and Geometric Perspectives
In this video, we explore the fundamental concept of the span of a set of vectors, a key topic in linear algebra. You'll learn how to determine whether a vector lies in the span of other vectors and what span means both algebraically and geometrically.

What You'll Learn:
• The formal definition of span
• How to determine if a vector is in the span of a set
• The geometric interpretation of span in R^2, R^3, and R^4
• Algebraic techniques to test for span
• Example problems to build conceptual and computational understanding

This lesson builds intuition for working with vector combinations and lays the groundwork for more advanced topics like linear independence and basis.
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Matrix Equations in Linear Algebra | Understanding and Solving Ax = b
In this video from our Linear Algebra series, we explore how to write and solve matrix equations, focusing on the foundational equation Ax = b. You’ll learn how linear systems can be expressed in matrix form and how to apply different methods to solve them effectively.

What You'll Learn:
• How to construct and interpret matrix-vector products
• When a matrix-vector product is defined based on dimensions
• Two methods of multiplication:

• The Linear Combination of Columns Method

• The Row Vector Rule (Dot Product Method)
  • Key properties of matrix-vector products
  • How to solve matrix equations using augmented matrices
  • The Equivalence Theorem and what it tells us about the solutions of Ax = b

We’ll walk through examples to illustrate how to determine matrix dimensions, assess whether solutions exist, and represent them in vector form.
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