System of Linear Equations
55 min 7 Examples
- What is Linear Algebra? What is a Matrix? and What is a Linear Equation?
- Determining whether an equation is Linear (Example #1)
- Definition of Consistent and Inconsistent Systems and Solution Types
- Determining the type of solutions for a system of 3 variables graphically (Example #2)
- Overview of Matrix Notation and Coefficient and Augmented Matrices
- Writing a Coefficient and Augmented Matrix given a Linear System (Examples #3-5)
- Finding a System of Equations given an Augmented Matrix (Example #6)
- Overview of Equivalent Systems and Equivalent Matrices
- Existence and Uniqueness for a Linear System (Example #7)
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
- Solving a system using Linear Combinations and RREF (Examples #1-5)
- Existence and Uniqueness Theorem for Row Reduction and Echelon Forms
- Existence and Uniqueness Questions for Row Reduction and Echelon Forms (Examples #6-9)
Vector Equations for Matrix Algebra
1 hr 27 min 9 Examples
- Representing a Vector as a Column Matrix or Column Vector; Algebraic Properties for Column Vectors
- Linear Combination and Span of Vectors: Definitions
- Express as a Linear Combination: Examples #1-3
- Determine if a given vector is a linear combination of the others (Example #4)
- Writing a System of Equations given a Vector Equation: Example
- Values making a vector in the plane generated by other vectors: Example
- Vector Equations, Linear Combination, and Span: Foundational Questions #1-3
The Matrix Equation Ax=b
57 min 7 Examples
- The Matrix Equation as matrix-vector multiplication; Writing in matrix-vector form: Example
- The Matrix Equation Theorem; Is the matrix equation consistent? (Example)
- Existence of Solutions Theorem
- Describing the solution of the matrix equation (Example #1)
- Determining if vectors span (Example #2)
- Determining if the columns of the matrix span: Examples #3-4
- Solve the Matrix Equation (Example #5)
Solution Sets of Linear Systems
1 hr 20 min 6 Examples
- Homogeneous Linear System, Trivial and Nontrivial Solutions: Definition
- Determining Nontrivial Solution and graphical Representation (Examples #1-2)
- Quick Review of vector basics: writing, finding, and graphing
- Parametric Vector Form: Overview, Graphically and Analytically
- Writing Solution Sets in Parametric Vector Form: Steps and Example
- Describing all solutions in Parametric Vector Form (Examples #1-2)
- Find an equation of the line through a parallel to b (Example #3)
- Writing a solution in both General and Parametric Vector Form (Example #4)
Linear Independence
1 hr 3 min 15 Examples
- Overview of Linear Independence; Facts, Definitions, and Theorems for Linear Independence
- Determining if vectors are linearly independent: Examples A-J
- Finding values in the span and making the set of vectors linearly dependent: Examples
- Existence and Uniqueness Theory Questions (T/F): Examples #1-6
Linear Transformations
1 hr 44 min 14 Examples
- Overview of Linear Transformations, Mapping, Domain, Codomain, Range, and the Matrix Transformation
- Find a vector x whose image under T is b (Examples #1-2)
- Is b in the range of the linear transformation? (Example #3)
- The Matrix of a Linear Transformation; Definition of the Standard Matrix; Five Basic Standard Matrix Transformations: Overview
- Describe geometrically what the Transformation does to each of the four vectors (Examples A-D)
- Finding the Standard Matrix: Four Examples
- Show that T is a Linear Transformation; Onto and One-to-One Mapping: Definition and Theorem
- Determine if the Linear Transformation is One-to-One and/or Onto (Examples #1-3)
- Prove whether T is a linear transformation (Examples #1-3)
Applications of Linear Systems
1 hr 35 min 4 Examples
- Applications of Linear Systems and Linear Models: Overview
- Economics Application: Overview and Example
- Nutrition and Diet Application: Example
- Network Flow: Overview and Examples #1-2
- Electrical Network Flow – Kirchhoff’s Law: Overview and Examples #1-2
Chapter Test
1 hr 48 min 15 Problems
- Solve the system using elementary row operations (Problem #1)
- Solve the system and write the answer in parametric vector form (Problem #2)
- Is b a linear combination fo vectors u and v? (Problem #3)
- Find scalars such that be is a linear combination (Problem #4)
- Match the transformation to the geometric description (Problem #5)
- Write the standard matrix for T given the transformation (Problem #6)
- By inspection determine if the set of vectors are linearly independent (Problem #7a-c)
- Find the image of the transformation (Problem #8)
- Determine if b is in the span (Problem #9)
- Determine if the transformation is one-to-one or onto (Problem #10)
- Determine the loop currents (Problem #11)
- Determine the general flow pattern (Problem #12)
- Use a migration matrix to estimate the population two years later (Problem #13)
- Prove T is a linear transformation (Problem #14)
- True or False (Problem #15a-d)