Linear Algebra and Geometry 1
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.
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Our Verdict
This course offers one of the most thorough treatments of linear algebra and geometry available on Udemy. Its impressive 47 hours of video content cover an extensive range of topics, providing a comprehensive foundation for understanding these critical areas. With over 175 solved problems and extensive instructor feedback throughout, students can feel confident that they will emerge with both theoretical knowledge and practical skills. While the sheer volume of material may present challenges to some learners, this course's unique value proposition lies in its commitment to delivering a truly conceptual understanding rather than focusing solely on rote memorization or mechanical problem-solving.\n\nDespite minor quibbles regarding certain section divisions and levels of theoretical rigor, the overall strength of this course is apparent. Expert instruction—anchored by consistent, clear language and precise presentation—elevates the program above its competitors, making it a prime resource for anyone aiming to deepen their appreciation and mastery of linear algebra and geometry concepts.
What We Liked
- Covers a broad range of linear algebra / geometry topics with 175 solved problems
- Detailed explanations focus on conceptual understanding, not just mechanics
- Instructor provides thorough and complete answers to course questions
- Well-structured organization & presentation approach supports learning
Potential Drawbacks
- Linear transformations section could be broken up further for easier digestion
- Notable volume of content may overwhelm learners unaccustomed to comprehensive math courses
- Some learners might require a more theorem-heavy presentation style