**Linear Equations**

## System of Linear Equations

55 min 7 Examples

- What is Linear Algebra? What is a Matrix? and What is a Linear Equation?
- Example of determining whether an equation is Linear
- Definition of Consistent and Inconsistent Systems and Solution Types
- Example of how to determine the type of solutions for a system of 3 variables graphically
- Overview of Matrix Notation and Coefficient and Augmented Matrices
- Example #1 Writing a Coefficient and Augmented Matrix given a Linear System
- Example #2 Writing a Coefficient and Augmented Matrix given a Linear System
- Example #3 Writing a Coefficient and Augmented Matrix given a Linear Systems
- Example of how to find a System of Equations given an Augmented Matrix
- Overview of Equivalent Systems and Equivalent Matrices
- Example of Existence and Uniqueness for a Linear System

## Reduced Row Echelon Form

2 hr 33 min 9 Examples

- How do we solve a system of linear equations?
- Understanding and Importance of the Identity Matrix
- Understanding Row Echelon Form and Reduced Row Echelon Form
- What is a Pivot Position and a Pivot Column?
- Steps and Rules for performing the Row Reduction Algorithm
- Example #1 Solving a system using Linear Combinations and RREF
- Example #2 Solving a system using REF
- Example #3 Solving a system using RREF
- Example #4 Solving a system using RREF
- Example #5 Solving a system using RREF
- Existence and Uniqueness Theorem for Row Reduction and Echelon Forms
- Existence and Uniqueness Question #1 Row Reduction and Echelon Forms
- Existence and Uniqueness Question #2 for Row Reduction and Echelon Forms
- Existence and Uniqueness Question #3 for Row Reduction and Echelon Forms
- Existence and Uniqueness Question #4 for Row Reduction and Echelon Forms

## Vector Equations for Matrix Algebra

1 hr 27 min 9 Examples

- How we represent a Vector as a Column Matrix or Column Vector
- Algebraic Properties for Column Vectors
- Definition of Linear Combination in terms of Vectors
- Definition of the Span of Vectors
- Example #1 Express as a Linear Combination
- Example #2 Express as a Linear Combination
- Example #3 Express as a Linear Combination
- Example #4 Determine if a given vector is a linear combination of the others
- Example of how to Write a System of Equations given a Vector Equation
- Example of what values will make a vector in the plane generated by other vectors
- Foundational Question #1 for Vector Equations, Linear Combination and the Span
- Foundational Question #2 for Vector Equations, Linear Combination and the Span
- Foundational Question #3 for Vector Equations, Linear Combination and the Span

## The Matrix Equation Ax=b

57 min 7 Examples

- The Matrix Equation viewed as matrix-vector multiplication
- Example of writing in matrix-vector form
- The Matrix Equation Theorem
- Example – Is the matrix equation consistent?
- Existence of Solutions Theorem
- Example #1 describing the solution of the matrix equation
- Example #2 determining if the vectors span?
- Example #3 determining if the columns of the matrix span
- Example #4 determining if the columns of the matrix span
- Example #5 solve the Matrix Equation

## Solution Sets of Linear Systems

1 hr 20 min 6 Examples

- What is a Homogeneous Linear System? Trivial and Nontrivial Solutions?
- Example #1 determining Nontrivial Solution and graphical Representation
- Example #2 determining Nontrivial Solution
- Quick Review of how to write/find/graph a vector (the basics)
- Overview of Parametric Vector Form Graphically and Analytically
- Steps for Writing Solution Sets in Parametric Vector Form with example
- Example #1 describe all solutions in Parametric Vector Form
- Example #2 describe all solutions in Parametric Vector Form
- Example #3 find an equation of the line through a parallel to b
- Example #4 writing a solution in both General and Parametric Vector Form

## Linear Independence

1 hr 3 min 15 Examples

- Overview of Linear Independence
- Facts, Definitions, and Theorems for Linear Independence
- Examples A-D for Determining if the vectors are linearly independent
- Examples E-F for Determining if the vectors are linearly independent
- Examples G-J for Determining if the vectors are linearly independent
- Example of finding values in the span
- Example of finding what values making the set of vectors linearly dependent
- Example #1 (T/F) Theory Question for Existence and Uniqueness
- Example #2 (T/F) Theory Question for Existence and Uniqueness
- Example #3 (T/F) Theory Question for Existence and Uniqueness
- Example #4 (T/F) Theory Question for Existence and Uniqueness
- Example #5 (T/F) Theory Question for Existence and Uniqueness
- Example #6 (T/F) Theory Question for Existence and Uniqueness

## Linear Transformations

1 hr 44 min 14 Examples

- Overview of Linear Transformations and the Matrix Transformation
- Definition of Mapping, Domain, Codomain, and Range and Understanding the Matrix Transformation
- Example #1 Find a vector x whose image under T is b
- Example #2 Find a vector x whose image under T is b
- Example #3 is b in the range of the linear transformation
- The Matrix of a Linear Transformation
- Definition of the Standard Matrix
- Overview of the Five Basic Standard Matrix Transformations
- Examples (A-D) Describe geometrically what the Transformation does to each of the four vectors
- Two Examples of Finding the Standard Matrix
- Two More Examples of Finding the Standard Matrix
- Show that T is a Linear Transformation
- Definition and Theorem for Onto and One-to-One Mapping
- Example #1 Determine if the Linear Transformation is One-to-One and/or Onto
- Example #2 Determine if the Linear Transformation is One-to-One and/or Onto
- Example #3 Determine if the Linear Transformation is One-to-One and/or Onto
- Steps for Proving a Linear Transformation
- Example #1 Prove whether T is a linear transformation
- Example #2 Prove whether T is a linear transformation
- Example #3 Show why T is a not a linear transformation

## Applications of Linear Systems

1 hr 35 min 4 Examples

- Applications of Linear Systems and Linear Models Overview
- Economics Application Overview
- Economics Application Example
- Nutrition and Diet Application Example
- Network Flow Overview
- Network Flow Example #1
- Network Flow Example #2
- Electrical Network Flow – Kirchhoff’s Law Overview
- Electrical Network Flow Example #1
- Electrical Network Flow Example #2