Linear Algebra and Geometry 2
Much more about matrices; abstract vector spaces and their bases; linear transformations animated with MANIM
What you will learn
How to solve problems in linear algebra and geometry (illustrated with 153 solved problems) and why these methods work.
Important concepts concerning vector spaces, such as basis, dimension, coordinates, and subspaces.
Linear combinations, linear dependence and independence in various vector spaces, and how to interpret them geometrically in R^2 and R^3.
How to recalculate coordinates from one basis to another, both with help of transition matrices and by solving systems of equations.
Row space, columns space and nullspace for matrices, and about usage of these concepts for solving various types of problems.
Linear transformations: different ways of looking at them (as matrix transformations, as transformations preserving linear combinations).
How to compose linear transformations and how to compute their standard matrices in different bases; compute the kernel and the image for transformations.
Understand the connection between matrices and linear transformations, and see various concepts in accordance with this connection.
Work with various geometrical transformations in R^2 and R^3, be able to compute their matrices and explain how these transformations work.
Understand the concept of isometry and be able to give some examples, and formulate their connection with orthogonal matrices.
Transform any given basis for a subspace of R^n into an orthonormal basis of the same subspace with help of Gram-Schmidt Process.
Compute eigenvalues, eigenvectors, and eigenspaces for a given matrix, and give geometrical interpretations of these concepts.
Determine whether a given matrix is diagonalizable or not, and perform its diagonalization if it is.
Understand the relationship between diagonalizability and dimensions of eigenspaces for a matrix.
Use diagonalization for problem solving involving computing the powers of square matrices, and motivate why this method works.
Be able to formulate and use The Invertible Matrix Theorem and recognise the situations which are suitable for the determinant test (and which are not).
Use Wronskian to determine whether a set of smooth functions is linearly independent or not; be able to compute Vandermonde determinant.
Work with various vector spaces, for example with R^n, the space of all n-by-m matrices, the space of polynomials, the space of smooth functions.
Why take this course?
🎓 Course Title: Linear Algebra and Geometry 2 with Dr. Hania Uscka-Wehlou
🚀 Headline: Dive Deeper into Matrices, Abstract Vector Spaces, and Linear Transformations with Animated Visualization using MANIM!
Course Description:
Welcome to "Linear Algebra and Geometry 2," an enriching journey through the world of linear algebra where we delve much deeper into the mysteries of matrices, abstract vector spaces, and linear transformations. This course is designed to challenge your understanding and provide you with a robust grasp of the concepts, complete with animated visualizations that bring abstractions to life.
Chapter 1: Abstract Vector Spaces and Related Stuff
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Introduction to the Course 📈
- Get acquainted with the course structure, objectives, and what you can expect to learn.
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Real Vector Spaces and Their Subspaces 🔬
- Explore the foundational axioms of vector spaces and master the art of identifying subspaces within them.
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Linear Combinations and Linear Independence ⚫️
- Understand linear combinations, learn to distinguish between dependent and independent sets, and apply Gaussian elimination for independence checks.
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Coordinates, Basis, and Dimension 🧮
- Dive into the concept of bases, coordinate systems, and the dimension of a vector space. Learn to determine if a set is a basis using the determinant test.
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Change of Basis ↔️
- Discover how to transition between different coordinate systems, solve for coordinates in new bases, and uncover the geometry behind these transformations.
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Row Space, Column Space, and Nullspace of a Matrix 📊
- Learn about the components of a matrix and how to find their respective spaces, including nullspaces, row spaces, and column spaces.
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Rank, Nullity, and Fundamental Matrix Spaces 🔄
- Understand the significance of rank, nullity, and their connection to the four fundamental matrix spaces.
Chapter 2: Linear Transformations
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Matrix Transformations from R^n to R^m 🔁
- Master the mapping between linear transformations and matrices, understand kernels, images, and inverse operators.
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Geometry of Matrix Transformations on R^2 and R^3 🌱
- Visualize linear transformations in two and three dimensions, exploring rotations, symmetries, projections, and their mathematical representations.
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Properties of Matrix Transformations 🔍
- Examine how linear transformations behave under various operations and transformations.
Chapter 3: Eigenvalues and Eigenvectors 🎲
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Eigenvalues and Eigenvectors 🌀
- Learn to compute eigenvalues and eigenvectors, understand their geometric significance, and explore eigenspaces.
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Diagonalization ✨
- Determine the diagonalizability of matrices and apply these techniques to solve problems involving matrix powers.
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Wrap-up Linear Algebra and Geometry 2 🏆
- Get a sneak peek at what you'll learn in the subsequent course, ensuring you're prepared for the future of linear algebra.
Extras 🎬
- Additional Courses and Future Plans 🚀
- Explore all the courses we offer and get a glimpse into our exciting plans for upcoming courses with hypothetical release dates.
Course Resources:
- Detailed Course Description 📋
- Access a comprehensive guide, including a list of all videos and problems solved throughout the course, in the resource file "001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_2.pdf". This document is referenced starting from video 1 ("Introduction to the course").
📚 Essential Note:
Remember that the content covered in this course can vary by institution and even year to year. Always confirm with your professor which parts of Linear Algebra and Geometry 2 are relevant to your exams and studies.
Embark on this mathematical adventure with "Linear Algebra and Geometry 2" and unlock the power of linear algebra for life! 🌟
[Enroll now and transform your understanding of Linear Algebra and Geometry with Dr. Hania Uscka-Wehlou's expert guidance and engaging content.]
Screenshots
Our review
🌟 Global Course Rating: 4.78 Recent Reviews Summary:
**Pros:**
- Engaging Content: The lectures are engaging, well-paced for accumulative learning, and the course structure is conducive to deep understanding.
- Comprehensive Explanations: Hania provides excellent explanations, illustrations, and numerous solved problems that help visualize complex concepts.
- In-Depth Coverage: The course offers an in-depth explanation of topics with lots of exercises, ensuring a thorough understanding of linear algebra.
- Highly Recommended: Many reviewers found the course to be highly beneficial, even stating it's the best way to learn linear algebra efficiently.
- Clear and Logical Presentation: The course is structured in a clear and logical manner, making learning Linear Algebra fun and rewarding.
- Diverse Learning Approaches: Concepts are explained from different angles, catering to different learning styles and deepening understanding.
- Quality of Instruction: Dr. Hania Wehlou is highly praised for her teaching methods, clarity, and patience in explaining concepts.
- Value for Money: Reviewers consider the course worth the investment, noting the abundance of exercises and theoretical depth covered.
**Cons:**
- Language Nuances: Some reviewers mentioned that language nuances or automatic voice-to-text transcription issues might arise, but this did not significantly impact the learning experience.
- Requires Dedication: The course requires dedication from students to get the most out of it, as it is university-level material.
Reviewer Experiences:
- One student reported a significant improvement in confidence and understanding after completing the course.
- Several reviewers highlighted the importance of following along with problems to maximize learning outcomes.
- A few mentioned that the subspaces section was particularly clear due to Hania's explanations.
- The course is praised for being systematic, providing context, and making abstract concepts accessible.
- Reviewers recommend this series to anyone interested in linear algebra, especially those looking to apply it to fields like machine learning.
Overall Impression: The Linear Algebra course by Dr. Hania Wehlou receives high praise from students for its comprehensive approach to teaching the subject. The course material is well-organized and covers a wide range of topics within linear algebra. The instructional style is clear, detailed, and patient, making it an excellent resource for learners at all levels who are serious about mastering linear algebra concepts. The positive reviews and high ratings indicate that this course stands out as an exceptional learning tool in the field of mathematics.