Linear Algebra and Geometry 1
Systems of equations, matrices, determinants, vectors, and geometry of straight lines and planes in the 3-space
What you will learn
How to solve problems in linear algebra and geometry (illustrated with 175 solved problems) and why these methods work.
Solve systems of linear equations with help of Gauss-Jordan or Gaussian elimination, the latter followed by back-substitution.
Interpret geometrically solution sets of systems of linear equations by analysing their RREF matrix (row equivalent with the augmented matrix of the system).
Matrix operations (addition, scaling, multiplication), how they are defined, how they are applied, and what computational rules hold for them.
Matrix inverse: determine whether a matrix is invertible; compute its inverse: both with (Jacobi) algorithm and by the explicit formula; matrix equations.
Determinants, their definition, properties, and different ways of computing them; determinant equations; Cramer's rule for n-by-n systems of equations.
Vectors, their coordinates and norm; geometrical vectors and abstract vectors, their addition and scaling: arithmetically and geometrically (in 2D and 3D).
Vector products (scaling, dot product, cross product, scalar triple product), their properties and applications; orthogonal projection and vector decomposition.
Analytical geometry in the 3-space: different ways of describing lines and planes, with applications in problem solving.
Compute distances between points, planes and lines in the 3-space, both by using orthogonal projections and by geometrical reasoning.
Determine whether lines and planes are parallel, and compute the angles between them (using dot product and directional or normal vectors) if they intersect.
How to geometrically interpret n-by-2 and n-by-3 systems of equations and their solution sets as intersection sets between lines in 2D or planes in 3D.
Understand the connection between systems of linear equations and matrix multiplication.
Invertible Matrix Theorem and its applications; apply determinant test in various situations.
Why take this course?
๐ Linear Algebra and Geometry 1: A Journey Through Linear Systems ๐
Course Overview
Unlock the Mysteries of Linear Algebra and Geometry in a 3-Space Dimension
Welcome to "Linear Algebra and Geometry 1" โ your comprehensive guide to mastering systems of equations, matrices, determinants, vectors, and the geometry of lines and planes. Whether you're a student aiming to enhance your mathematical skills or a professional seeking to apply these concepts in natural sciences, this course is designed to cater to your needs.
Chapter 1: Systems of Linear Equations ๐
S1. Introduction to the Course ๐
- Get acquainted with the course structure and objectives.
S2. Some Basic Concepts ๐
- Refresh your memory on key concepts that will be the foundation of our journey.
S3. Systems of Linear Equations; Building up Your Geometrical Intuition ๐
- Understand the basics of linear equations and explore the geometric interpretation they offer.
S4. Solving Systems of Linear Equations; Gaussian Elimination ๐งฎ
- Master the Gaussian elimination method and learn about back-substitution and Gauss--Jordan elimination for various types of solutions.
S5. Some Applications in Mathematics and Natural Sciences ๐
- Discover real-world applications that bring linear equations to life.
Chapter 2: Matrices and Determinants โซ๏ธ
S6. Matrices and Matrix Operations ๐
- Dive into the world of matrices, their operations, and different types of matrices.
S7. Inverses; Algebraic Properties of Matrices ๐
- Unravel the concept of matrix inverses and explore the algebraic properties that make them essential.
S8. Determinants and Their Significance ๐
- Learn how determinants can solve problems and provide insights into systems of linear equations.
Chapter 3: Analytical Geometry of Lines and Planes ๐
S9. Lines in R^2 ๐ฃ๏ธ
- Explore various representations of lines in the plane and how to compute their equations.
S10. Planes in R^3 ๐ชจ
- Delve into the complexities of representing and describing planes in three-dimensional space.
S11. Lines in R^3 ๐ซ
- Investigate lines in 3D space, their equations, and how they intersect with other geometrical figures.
S12. Geometry of Linear Systems; Incidence Between Lines and Planes โซ
- Learn to compute the intersections of lines and planes and understand their significance.
S13. Distance Between Points, Lines, and Planes ๐
- Discover methods to calculate distances between various points and geometrical figures.
Chapter 4: Special Topics and Future Directions ๐
S14. Some Words About the Next Course ๐
- Get a sneak peek at what you'll encounter in our subsequent course on Linear Algebra and Geometry.
S15. Extras ๐
- Explore additional resources, future course plans, and more to enhance your learning experience.
Course Materials and Resources ๐
- Detailed Course Outline: A comprehensive guide with all video titles and problem solutions. Available in the resource file "001 Outline_Linear_Algebra_and_Geometry_1.pdf".
- Final Exam Preparation: Ensure you're well-prepared for your final exam by checking with your professor which parts of the course are required.
Note: The content and schedule for future courses are subject to change, so stay tuned for updates!
Embark on this mathematical adventure with confidence, as we guide you through the intricacies of linear algebra and geometry in a 3-space dimension. With our expert instructor and a wealth of resources at your fingertips, you're set for success! ๐โจ
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Our review
Course Review for Linear Algebra
Overview: The Linear Algebra course has been met with high praise from recent reviewers, boasting an impressive global course rating of 4.90. The majority of the reviews are overwhelmingly positive, with several learners expressing that it is the best conceptual learning experience they have had. The course stands out for its comprehensive nature, pedagogical resources like examples in Geogebra, and the deep understanding of geometry and computations it imparts.
Pros:
- Comprehensive Curriculum: The course covers a wide range of topics within linear algebra, ensuring learners get a full understanding of the subject matter.
- Pedagogical Resources: With examples in Geogebra and other visual aids, the course is designed to cater to different learning styles and enhance comprehension.
- High-Quality Instruction: The instructor's presentation skills are praised for being clear, well-structured, and engaging, making even complex concepts accessible.
- Real World Application: The course does not just focus on theoretical aspects but also shows how linear algebra can be applied in various contexts.
- Clear Explanations: The explanations provided are detailed and logical, helping learners to visualize and understand the concepts better.
- High Production Quality: The video quality, presentation style, and organization of the content are all commended for their professionalism.
- Language Proficiency: Although the instructor is not a native English speaker, the academic English used is described as crystal clear.
- Inclusive and Encouraging: The instructor's approach is encouraged and patient, making learners feel supported throughout the course.
- Variety of Learning Materials: The inclusion of additional graphics and Q&A sessions enhance the learning experience.
- Value for Money: Several learners indicate that they intend to contribute full price eventually, highlighting the perceived value of the course.
Cons:
- Advanced Preparation Required: It is recommended that learners have a high school understanding of mathematics to fully benefit from the course.
- Pace May Be Slower for Some: The gentle pace may be too slow for learners who are already familiar with the subject or those looking for a faster-paced learning environment.
- Limited Focus on Challenging Concepts: While some learners appreciate the gradual introduction of more challenging concepts, others may wish for a broader range of difficult topics right from the start.
- Additional Solved Problems Desired: Some learners express a desire for more solved problems to be included in the course.
Conclusion: The Linear Algebra course is an exceptional resource for learners who are new to the subject or looking to deepen their understanding of linear algebra. The course's blend of comprehensive content, clear explanations, and engaging instruction make it stand out as a top-rated learning experience in this field. While it is well-suited for those with a solid foundation in high school mathematics, it may not cater to those seeking a rapid pace or an introduction to more advanced topics immediately. Overall, the course is highly recommended for its quality of content and instruction.
Learner Testimonials:
- "This course has been the best conceptual learning experience I've had on this platform. The instructor explains concepts with such clarity that even someone starting from scratch can grasp them."
- "The course is not just theoretical; it shows how linear algebra applies to real-world problems, which I found incredibly useful."
- "Dr. Hania knows her content exceptionally well and presents it in a very meticulous way that's easy to follow."
- "As a learner who's coming back to linear algebra after some time, the pace of this course is perfect. It allows me to fully understand each concept before moving on."
- "The additional graphics and practice questions are a great help in reinforcing my understanding of the material."
- "I wish there were more courses by Dr. Hania; she's an excellent teacher!"
Final Takeaway: This Linear Algebra course is a testament to effective online learning, with its comprehensive curriculum and engaging instruction. It's an invaluable resource for anyone interested in mastering linear algebra or deepening their understanding of its applications. The positive feedback from learners underscores the quality and effectiveness of the course as a whole.