Linear Algebra
part 3 of 3

A total of 50 hours of lectures

This is an academic level course for university and college engineering. Due to its size, it is divided into three parts. This page describes the third of those three parts.

Level - Intermediate

  • High-school and college mathematics (mainly arithmetics, some trigonometry, polynomials)
  • Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
  • Linear Algebra and Geometry 2 (vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors, diagonalization)
  • Some basic calculus
    Basic knowledge of complex numbers (this course contains a short introduction to complex numbers)

Curriculum

Make sure that you check with your professor what parts of the course you will need for your exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

Linear Algebra, part 3 of 3

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Get the outline

A detailed list of all the lectures in part 3 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.

Get Linear Algebra part 3 on Udemy

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Course Objectives & Outcomes for part 3

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How to solve problems in linear algebra and geometry (illustrated with 144 solved problems) and why these methods work.
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Use diagonalization of matrices for solving various problems from different branches of mathematics (ODE, dynamical systems).
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Work with geometric concepts as length (norm), distance, angles, and orthogonality in non-geometric setups.
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Orthogonal and orthonormal bases, and Gram-Schmidt process in various inner product spaces.
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Symmetric matrices and their properties; orthogonal diagonalization: how it is done and how to understand it geometrically.
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Quadratic forms and their connection to symmetric matrices: uniqueness of this correspondence and its consequences.
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Some concepts from abstract algebra: group, ring, field, and isomorphism; understand the concept of isomorphic vector spaces.
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Note: all the vector spaces discussed in this course are spaces over R (not over the field of complex numbers), and all our matrices have only real entries.
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Solve more advanced problems on eigendecomposition and orthogonality than in the second course.
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Inner product spaces different from R^n: space of continuous functions, spaces of polynomials, spaces of matrices.
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Pythagorean Theorem, Cauchy-Schwarz inequality, and triangle inequality in various inner product spaces.
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Min-max problems using Cauchy-Schwarz inequality, Best Approximation Theorem, least squares solutions.
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Positive/negative definite matrices, indefinite matrices; various methods of determining definiteness of matrices.
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Geometry of quadratic forms in two and three variables: conic sections and quadratic surfaces.
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Crowning of the course and a natural consequence of all the other topics: Singular Value Decomposition and pseudoinverses.

Andrzej P.

Udemy student

Incredible lectures! So many topics covered in such a clear and structured manner. The exercises were especially helpful in understanding the theory and how to use it. Really helpful. Highly recommend these lectures.

Tetyana M.

Lecturer mathematics

The present is a logical continuation of two earlier courses by Hania, and it is presented in the same spirit – extremely carefully and in detail. It bridges elementary linear algebra with its most standard mathematical applications, motivating the students to further study both. Noteworthy here are applications under the heading of singular value decomposition, basic for what may be called linear data analysis, key to engineering and statistical computations. A warmly recommended course, for prospective data analysts not least.

Murat F.

Udemy student

Best courses I took about the linear algebra (including the ones I took at the school). Anyone interested in topics should definitely enroll these courses (the Linear Algebra series by Prof Hania).

As an engineering student these lectures made me interested even in the more theoretical parts. I’ll definitely learn more about linear algebra, since these courses gave me great foundations.

The instructor is very knowledgeable and polite (gives great feedback even to my most stupid questions 🙂 ). I thank both instructors for their great effort on these courses, and look forward to their next courses!

Derick B.

Udemy student

I am pretty thankful for COVID-19 as it allowed professor Hania to be able to explore the idea of using tech to record high-quality lectures in mathematics: a crucial subject in pretty much everything we do. I love maths since school but the level of rigor was lacking. Even though I could get the picture of the mathematical idea, I felt something that kept bumping up in my head and bothering me. I revised lots of books. But at some point, the notation and level of detail became a nightmare to me. Then, I decided to go for free online courses. I found MIT OpenCourseWare. First, I took chemistry. I loved it as I could understand nearly everything. Then, I enrolled at Single Variable Calculus, but I felt at some level the rigor and exercises were simple. Finally, I took Multivariable Calculus. The course was great. I did understand the concepts and did a lot of exercises. Nonetheless, when I grabbed an MVC book the concepts contained a lot of notation and mathematical details I could not get my head around. Furthermore, if I got stuck, I gave up even after doing lots of research because of the complexity and time issues …. I panicked. So I try to find more rigorous courses in mathematics. But this time in Linear Algebra with Professor Hania. Now, I am more confident in that sense. The lectures are also well structured and geometrical pictures are richer and more meaningful than ever. Feedback is also there at your disposal so you can feel you can master a topic you were doubtful about even before starting. Now, I appreciate maths in more depth than ever. Last but not least, I am no longer afraid of grabbing an advanced book and setting out to understand it. I know the dynamics of how mathematics works at a high level. Thanks, professor Hania from the bottom of my heart. Words are not enough to describe my gratitude.

Alex

Udemy student

Absolutely loved this course! I had also taken the previous two. Hania explains things in such detail showing both clear logical reasoning and geometrical intuition. She does a great job of only using terms/methods/proofs that she explained previously.

All knowledge builds upon previous content. It is very well structured with plenty of solved problems. She doesn’t overwhelm you with math symbols and instead focuses on concepts.

For any problem, I found it helpful to pause the video and solve it myself before resuming to see her formulate the solution.

This is a quality course that will give you a ground up understanding of linear algebra. I’m glad there are more videos to come!

Bernard X.

Udemy student

Parfait pour moi. Je pense avoir beaucoup appris avec beaucoup de plaisir.

Wanda W.

Udemy student

The third course is definitely harder than the first and the second, but many illustrations and examples make it easy to follow and understand the content anyway. I have seen about 20 percent by now; very excited and curious about SVD at the end.

Terrence L.

Udemy student

Professor Hania keeps me wanting to learn more from her! Great Experience!

Richard B.

Lecturer mathematics

This extends Hania’s courses one and two with the same title, and it should be the most useful in terms of applications. It also is the hardest to present, if to follow the explicit style of the predecessors. Some abstraction is inevitable, handled carefully enough, but time is mostly spent explaining ideas and examining examples. The lectures are as easy to follow as any lectures could be, perhaps deceptively so, for some ideas are subtle. The students willing to listen and spend some time with the problems will be richly rewarded.

Andrzej P.

Udemy student

Incredible lectures! So many topics covered in such a clear and structured manner. The exercises were especially helpful in understanding the theory and how to use it. Really helpful. Highly recommend these lectures.

Tetyana M.

Lecturer mathematics

The present is a logical continuation of two earlier courses by Hania, and it is presented in the same spirit – extremely carefully and in detail. It bridges elementary linear algebra with its most standard mathematical applications, motivating the students to further study both. Noteworthy here are applications under the heading of singular value decomposition, basic for what may be called linear data analysis, key to engineering and statistical computations. A warmly recommended course, for prospective data analysts not least.

Murat F.

Udemy student

Best courses I took about the linear algebra (including the ones I took at the school). Anyone interested in topics should definitely enroll these courses (the Linear Algebra series by Prof Hania).

As an engineering student these lectures made me interested even in the more theoretical parts. I’ll definitely learn more about linear algebra, since these courses gave me great foundations.

The instructor is very knowledgeable and polite (gives great feedback even to my most stupid questions 🙂 ). I thank both instructors for their great effort on these courses, and look forward to their next courses!

Derick B.

Udemy student

I am pretty thankful for COVID-19 as it allowed professor Hania to be able to explore the idea of using tech to record high-quality lectures in mathematics: a crucial subject in pretty much everything we do. I love maths since school but the level of rigor was lacking. Even though I could get the picture of the mathematical idea, I felt something that kept bumping up in my head and bothering me. I revised lots of books. But at some point, the notation and level of detail became a nightmare to me. Then, I decided to go for free online courses. I found MIT OpenCourseWare. First, I took chemistry. I loved it as I could understand nearly everything. Then, I enrolled at Single Variable Calculus, but I felt at some level the rigor and exercises were simple. Finally, I took Multivariable Calculus. The course was great. I did understand the concepts and did a lot of exercises. Nonetheless, when I grabbed an MVC book the concepts contained a lot of notation and mathematical details I could not get my head around. Furthermore, if I got stuck, I gave up even after doing lots of research because of the complexity and time issues …. I panicked. So I try to find more rigorous courses in mathematics. But this time in Linear Algebra with Professor Hania. Now, I am more confident in that sense. The lectures are also well structured and geometrical pictures are richer and more meaningful than ever. Feedback is also there at your disposal so you can feel you can master a topic you were doubtful about even before starting. Now, I appreciate maths in more depth than ever. Last but not least, I am no longer afraid of grabbing an advanced book and setting out to understand it. I know the dynamics of how mathematics works at a high level. Thanks, professor Hania from the bottom of my heart. Words are not enough to describe my gratitude.