Matrix Algebra with Computational Applications

Matrix Algebra with Computational Applications is a collection of Open Educational Resource (OER) materials designed to introduce students to the use of Linear Algebra to solve real world problems. These materials were developed specifically for students and instructors working in a “flipped classroom” model that emphasizes hands-on problem solving activities during class meetings, with students watching lectures and completing readings and assignments outside of the classroom.

Linear Algebra Mathematics

Author: Dirk Colbry
Publisher: Michigan State University
Source: LibreTexts
License: CC BY-NC
Language: EN
Category: Basic Sciences

Read online Download PDF (open licence)

Contents· 253 sections

01: Matrix Algebra class preparation checklist
1.1: Get Jupyter Working
1.2: Review Python Packages
1.3: Complete Pre-class Assignment
02: 01 In-Class Assignment - Welcome to Matrix Algebra with computational applications
2.0: Introduction
2.1: Class Procedures
2.2: Introductions
2.3: Example
2.4: Syllabus, Schedule and other Procedures
2.5: Download and review next pre-class assignment
03: 02 Pre-Class Assignment - Vectors
3.0: Introduction
3.1: Introducing the Course Textbooks
3.2: Today’s Reading
3.3: Scalars, Vector and Tensors
3.4: Assignment wrap-up
04: 02 In-Class Assignment - Vectors
4.0: Introduction
4.1: Example linear system
4.2: Pre-class assignment review
4.3: Vectors in Python
05: 03 Pre-Class Assignment - Linear Equations
5.0: Introduction
5.1: System of Linear Equations
5.2: Visualizing the problem
5.3: Multidimentional Spaces
5.4: Matrix Notation
5.5: Assignment wrap-up
06: 03 In-Class Assignment - Solving Linear Systems of equations
6.0: Introduction
6.1: Pre-class assignment review
6.2: Jacobi Method for solving Linear Equations
6.3: Numerical Error
07: 04 Pre-Class Assignment - Python Linear Algebra Packages
7.0: Introduction
7.1: The Syntax for Systems of Linear Equations
7.2: Introduction to Gauss Jordan Elimination
7.3: Gauss Jordan Elimination and the Row Echelon Form
7.4: Gauss Jordan Practice
7.5: Assignment wrap up
08: 04 In-Class Assignment - Linear Algebra and Python
8.0: Introduction
8.1: Pre Class Review
8.2: Solving Systems of Linear Equations
8.3: Underdetermined Systems
8.4: Practice Curve Fitting Example
09: 05 Pre-Class Assignment - Gauss-Jordan Elimination
9.0: Introduction
9.1: Sympy RREF function
9.2: Calculating Vector Length, Normalization, Distance and Dot
9.3: Vector spaces in R^n
9.4: Assignment wrap up
10: 05 In-Class Assignment - Gauss-Jordan
10.0: Introduction
10.1: Pre-class assignment review
10.2: Generalize the procedure
10.3: Basic Gauss Jordan
11: 06 Pre-Class Assignment - Matrix Mechanics
11.0: Introduction
11.1: Dot Product Review
11.2: Matrix Multiply
11.3: Identity Matrix
11.4: Elementary Matrices
11.5: Solving Many Systems (at the same time)
11.6: Assignment wrap up
12: 06 In-Class Assignment - Matrix Multiply
12.0: Introduction
12.1: Review of Pre class assignment
12.2: Systems of Linear Equations with Many Solutions
12.3: Matrix Multiply
13: 07 Pre-Class Assignment - Transformation Matrix
13.0: Introduction
13.1: The Inverse Matrix (aka A^-1)
13.2: Transformation Matrix
13.3: Assignment wrap-up
14: 07 In-Class Assignment - Transformations
14.0: Introduction
14.1: Review of Pre-Class Assignment
14.2: Affine Transforms
14.3: Fractals
15: 08 Pre-Class Assignment - Robotics and Reference Frames
15.0: Introduction
15.1: Review
15.2: 2D Forward Kinematics
15.3: Assignment wrap-up
16: 08 In-Class Assignment - The Kinematics of Robotics
16.0: Introduction
16.1: Review Pre-class Assignment
16.2: Pick and Place
16.3: Odd Clock
17: 09 Pre-Class Assignment - Determinants
17.0: Introduction
17.1: Introduction to Determinants
17.2: Properties of Determinants
17.3: One interpretation of determinants
17.4: Cramer’s Rule
17.5: Assignment wrap-up
18: 09 In-Class Assignment - Determinants
18.0: Introduction
18.1: Review Pre-class Assignment
18.2: Algorithm to calculate the determinant
18.3: Using Cramer’s rule to solve Ax=b
19: 10 Pre-Class Assignment - Eigenvectors and Eigenvalues
19.0: Introduction
19.1: Eigenvectors and Eigenvalues
19.2: Solving Eigenproblems - A 2x2 Example
19.3: Introduction to Markov Models
19.4: Assignment wrap-up
20: 10 In-Class Assignment - Eigenproblems
20.0: Introduction
20.1: Pre Class Review
20.2: Introduction to Markov Models
20.3: Another Markov Model Example
21: 11 Pre-Class Assignment - Vector Spaces
21.0: Introduction
21.1: Basis Vectors
21.2: Vector Spaces
21.3: Lots of Things Can Be Vector Spaces
21.4: Assignment wrap-up
22: 11 In-Class Assignment - Vector Spaces
22.0: Introduction
22.1: Review Pre-class Assignment
22.2: Introduction to subspaces
22.3: Basis Vectors
22.4: Vector Spaces
23: 12 Pre-Class Assignment - Matrix Spaces
23.0: Introduction
23.1: Review the Properties of Invertible Matrices
23.2: The Basis of a Vector Space
23.3: Change of Basis
23.4: Assignment wrap-up
24: 12 In-Class Assignment - Matrix Representation
24.0: Introduction
24.1: Review Pre-class assignment
24.2: Matrix Representation of Vector Spaces
24.3: Practice Nutrition
25: 13 Pre-Class Assignment - Projections
25.0: Introduction
25.1: Orthogonal and Orthonormal
25.2: Code Review
25.3: Gram-Schmidt
25.4: Assignment wrap-up
26: 13 In-Class Assignment - Projections
26.0: Introduction
26.1: Pre-class Review
26.2: Understanding Projections With Code
26.3: Gram-Schmidt Orthogonalization Process
27: 14 Pre-Class Assignment - Fundamental Spaces
27.0: Introduction
27.1: Orthogonal Complement
27.2: The Four Fundamental Spaces
27.3: Independent Learning
27.4: Assignment wrap-up
28: 14 In-Class Assignment - Fundamental Spaces
28.0: Introduction
28.1: Pre-class assignment review
28.2: Four Fundamental Subspaces
28.3: Practice Example
29: 15 Pre-Class Assignment - Diagonalization and Powers
29.0: Introduction
29.1: Eigenvalues and eigenvectors review
29.2: Diagonalizable Matrix
29.3: Assignment wrap-up
30: 15 In-Class Assignment - Diagonalization
30.0: Introduction
30.1: Pre-class Assignment Review
30.2: Diagonalization
30.3: The Power of a Matrix
31: 16 Pre-Class Assignment - Linear Dynamical Systems
31.0: Introduction
31.1: Linear Dynamical Systems
31.2: Markov Models
31.3: Ordinary Differential Equations
31.4: Assignment wrap up
32: 16 In-Class Assignment - Linear Dynamical Systems
32.0: Introduction
32.1: Epidemic Dynamics - Discrete Case
32.2: Epidemic Dynamics - Continuous Model
32.3: Population Dynamics
33: 17 Pre-Class Assignment - Decompositions
33.0: Introduction
33.1: Matrix Decomposition
33.2: Decompositions
33.3: Assignment wrap-up
34: 17 In-Class Assignment - Decompositions and Gaussian Elimination
34.0: Introduction
34.1: Pre-Class Review
34.2: Decompositions
34.3: Focus on LU
35: 18 Pre-Class Assignment - Inner Product
35.0: Introduction
35.1: Inner Products
35.2: Inner Product on Functions
35.3: Assignment wrap-up
36: 18 In-Class Assignment - Inner Products
36.0: Introduction
36.1: Pre-class Review
36.2: Minkowski Geometry
36.3: Function Approximation
37: 19 Pre-Class Assignment - Least Squares Fit (Regression)
37.0: Introduction
37.1: Least Squares Fit
37.2: Linear Regression
37.3: One-to-one and Inverse transform
37.4: Inverse of a Matrix
37.5: Assignment wrap-up
38: 19 In-Class Assignment - Least Squares Fit (LSF)
38.0: Introduction
38.1: LSF Pre-class Review
38.2: Finding the best solution in an overdetermined system
38.3: Pseudoinverse
39: 20 In-Class Assignment - Least Squares Fit (LSF)
39.0: Introduction
39.1: LSF Example - Tracking the Planets
39.2: LSF Example - Predator Pray Model
39.3: LSF Example - Estimating the best Ellipses
40: Pre-Class Assignment - Solve Linear Systems of Equations
40.0: Introduction
40.1: Linear Systems
40.2: Under Defined Systems
40.3: Invertible Systems
40.4: Overdefined systems
40.5: System Properties
40.6: Assignment wrap up
41: 21 In-Class Assignment - Solve Linear Systems of Equations using QR Decomposition
41.0: Introduction
41.1: Pre-Class Review
41.2: Solve Linear Systems
41.3: Overdetermined Systems
41.4: Underdetermined Systems
42: Supplemental Materials - Python Linear Algebra Packages
42.0: Introduction
42.1: Matplotlib
42.2: Review of Python Math Package
42.3: Review of Python Numpy Package
42.4: Advanced Python Indexing
42.5: LaTeX Math
42.6: Jupyter Tips and Tricks
43: Jupyter Getting Started Guide
43.1: Getting Jupyter Working
43.2: Example running python code in Jupyter Notebooks
43.3: Video review of Python, IPython, and IPython notebooks
43.4: Installing Anaconda Python
43.5: Introduction to the Engineering Jupyter Account
43.6: More Information
44: Python Linear Algebra Packages
44.0: Introduction
44.1: AnswerCheck tool
44.2: Review of Python Math Package
44.3: Review of Python Numpy Package
44.4: Advanced Python Indexing
44.5: LaTeX Math