Linear Algebra - Foundations to Frontiers
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Course Details
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edX online courses
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Harvard University, the Massachusetts Institute of Technology, and the
University of California, Berkeley, are just some of the schools that you have
at your fingertips with edX. Through massive open online courses (MOOCs) from
the world's best universities, you can develop your knowledge in literature,
math, history, food and nutrition, and more. These online classes are taught
by highly-regarded experts in the field. If you take a class on computer
science through Harvard, you may be taught by David J. Malan, a senior
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Course Description
Linear Algebra: Foundations to Frontiers (LAFF) is packed full of challenging, rewarding material that is essential for mathematicians, engineers, scientists, and anyone working with large datasets. Students appreciate our unique approach to teaching linear algebra because:
- It’s visual.
- It connects hand calculations, mathematical abstractions, and computer programming.
- It illustrates the development of mathematical theory.
- It’s applicable.
In this course, you will learn all the standard topics that are taught in typical undergraduate linear algebra courses all over the world, but using our unique method, you'll also get more! LAFF was developed following the syllabus of an introductory linear algebra course at The University of Texas at Austin taught by Professor Robert van de Geijn, an expert on high performance linear algebra libraries. Through short videos, exercises, visualizations, and programming assignme...
Linear Algebra: Foundations to Frontiers (LAFF) is packed full of challenging, rewarding material that is essential for mathematicians, engineers, scientists, and anyone working with large datasets. Students appreciate our unique approach to teaching linear algebra because:
- It’s visual.
- It connects hand calculations, mathematical abstractions, and computer programming.
- It illustrates the development of mathematical theory.
- It’s applicable.
In this course, you will learn all the standard topics that are taught in typical undergraduate linear algebra courses all over the world, but using our unique method, you'll also get more! LAFF was developed following the syllabus of an introductory linear algebra course at The University of Texas at Austin taught by Professor Robert van de Geijn, an expert on high performance linear algebra libraries. Through short videos, exercises, visualizations, and programming assignments, you will study Vector and Matrix Operations, Linear Transformations, Solving Systems of Equations, Vector Spaces, Linear Least-Squares, and Eigenvalues and Eigenvectors. In addition, you will get a glimpse of cutting edge research on the development of linear algebra libraries, which are used throughout computational science.
MATLAB licenses will be made available to the participants free of charge for the duration of the course.
We invite you to LAFF with us!
Sort By
Student
8/10 starsCompleted
4 years, 4 months ago
The LAFF course used an interesting approach to present linear/matrix algebra content to a wide audience. Having taken math up to multivariable calculus, I was able to complete all assignments of this course within 3 weeks and hope to gain exposure prior to take calculus-based linear algebra. I found this course did an overall OK job. However, I would like to critique the PROGRAMMING portion of this course -- the course is missing the opportunity of furthering the MATLAB (or IPython). A lot of exercises were very insightful, however, with poor directions. The lectures were also quite dry. In addition, I believe the instructors could make the course more challenging by revoking access to some of the homework answers, certainly for the 3 exams. The course can also be improved by better organization: the concepts covered don't seem to require a full 12-week.
Patricio Barletta
6/10 starsCompleted
4 years, 9 months ago
I find no flaw in the instructor, nor the provider; my review is about the content.
Now, if you are (wannabe) a CS student and/or have never been exposed to college-level linear algebra, then this course may prove useful. But, if you mainly care about learning linear algebra then this course does a poor job. In that case I highly recommend Gilbert Strang course @ MIT OCW.
I think the Professor made a sincere, and big, effort to develop a new way to teach linalg but, sadly, it just didn't work. Gilbert Strang teaches this same subject in a much more old fashioned way (him and the blackboard) and, god, what a great set of lessons.
- 58 reviews
- 58 completed
5 years, 2 months ago
Elemental algebraic structures and algorithms are presented with a very strong focus on computation and time/space optimization. The content is very solid and interesting per se but, sadly, the lecture style is boring, the chosen programming environment is cumbersome and the assignments are only a little more than grinding.
Student
10/10 starsCompleted
5 years, 3 months ago
I studied Linear Algebra as part of a Maths-Physiscs course some time ago. As a result of the teaching format in LAFF, I have a better understanding of the concepts. Video snippets followed by examples and exercises was more conducive to learning that an hour of reams of alpha-numerics on a whiteboard. I enjoyed the way you partition matrices when implementing algorithms. I just completed the matlab version but I am interested in looking over the iPython notebooks.
Robert's favourite saying: 'go do the homework and I'll see you in the next video'.
Aon duine ata ag iarriadh a chuid eolas matamataice a feabhsú, beir bua. Seo é an chursa..
Le gach dea-ghuI
Jim Coen
- 1 review
- 1 completed
5 years, 4 months ago
I've studied Linear Algebra before in my first year in college. However, this course here is something else. It really made me understand what Linear Algebra is; what are the meaning of the things I've always used blindly just to solve problems without really understanding what is going on. The contents are smoothly put, the instructor is very good, he just takes you step by step to master the topic. Although the course I've attended hadn't a serious grading, the homework helped me a lot to understand the materials and to understand not memorize the properties of special matrices for example. I recommend this course for anyone who didn't like Linear Algebra but realized its importance and cannot find the right place to learn it; this is what you're looking for.
Student
10/10 starsCompleted
5 years, 4 months ago
The content was excellent, particularly for anyone with a computer science tilt.
I particularly enjoyed two aspects of the course. The first was the focus on core principles that I had failed to learn previously taking linear algebra- two examples being that matrix-vector multiplication is *defined* as to represent a linear function / transformation and that matrix-matrix multiplication is *defined* as to represent linear function composition.
The second aspect was the focus on proof writing- specifically in proving the correctness of algorithms. This is really where the course stands out those with a focus on Comp Sci.
- 1 review
- 1 completed
5 years, 4 months ago
This was a great course! It goes into many relevant aspects of Linear Algebra and explains all the concepts very well. I loved the focus on algorithms and the MathLab homework. Also, I really liked how we explored the frontiers of Linear Algebra, making it feel like an interesting course both for people who haven't learned Linear Algebra yet as well as people who already have some experience.
- 1 review
- 1 completed
5 years, 4 months ago
好吧,我更想用中文写:
非常,非常感谢这门课程的提供者,如此无私地提供高质量的课程.
我的背景是生物,并且在我的本科中并没有学习线性代数. 现在研究生物信息,才发现统计中太经常使用线性代数,所以我需要从头学起. 而这门课程,提供了一个非常好的平台,和学习材料,让我有机会学会线性代数.
由于我的目的并不是矩阵编程,所以我跳过了很多编程作业,对不起.
课程中的enrichment部分,对于我来说,很多地方也很难,我觉得我还需要回头再去读.
我太喜欢这门课程了!所有的作业都让我觉得确实有必要,有的时候是课没有听懂,材料也没看懂,然后做了作业才突然明白的!而且老师很有心有用心地设置每一个题目,配有视频讲解.难以找到比这设计更合理的课程了.
也是通过这门课程,我发觉了edX平台的长处,及时的练习确实比看完所有的视频再去做作业要合理(coursera上是这样的模式).
非常,非常感谢老师们的无私奉献!
- 1 review
- 1 completed
5 years, 4 months ago
I did linear algebra longtime back in the college and also got exposure of the subject in econometrics and statistics course. But always I felt some gaps - this course provides a good link between usual approach plus programming intuition and why ? Stuff like Matrix Vector Multiplication, LU Decomposition and QR Decomposition , orthogonal projection etc.
The course has been developed with a lot effort and research, it starts with simple intuition and builds up the concept very well and every exercise has been chosen for a specific purpose - to introduce a concept in the latter stage. Another unique part of the concept mainly emphasized by Robert is "understanding" matrix and vector by slicing and dicing - which gives an extra edge in visualization, why is happening and why.
I would say anybody want to pursue econometrics , statistics and machine learning should start with this. After this course you will be a good position to p...
I did linear algebra longtime back in the college and also got exposure of the subject in econometrics and statistics course. But always I felt some gaps - this course provides a good link between usual approach plus programming intuition and why ? Stuff like Matrix Vector Multiplication, LU Decomposition and QR Decomposition , orthogonal projection etc.
The course has been developed with a lot effort and research, it starts with simple intuition and builds up the concept very well and every exercise has been chosen for a specific purpose - to introduce a concept in the latter stage. Another unique part of the concept mainly emphasized by Robert is "understanding" matrix and vector by slicing and dicing - which gives an extra edge in visualization, why is happening and why.
I would say anybody want to pursue econometrics , statistics and machine learning should start with this. After this course you will be a good position to pursue those subjects at ease.
Student
10/10 starsCompleted
5 years, 4 months ago
This is a Linear Algebra course that takes you - as promised - from foundations to frontiers. What you get from the course depends on whether your initial knowledge is closer to foundations or frontiers. I can only speculate - as I'm no beginner - that the course is very useful to every beginner who is at least slightly ambitious and self-disciplined. There are a lot of lessons with good explanations and tons of exercices. Some of them require a rigorous, more or less difficult proof - or a reasonable guess, but most are fairly simple and provide ample opportunity for training. Often the questions are designed to make you look for a pattern or a way to reuse a previous result.
If you're closer to the frontier, the course may look less interesting, as it is being approached very late and possibly never reached. (Well, that's by design!) At the beginning, depending on your country, you may feel that you know most from school, but ev...
This is a Linear Algebra course that takes you - as promised - from foundations to frontiers. What you get from the course depends on whether your initial knowledge is closer to foundations or frontiers. I can only speculate - as I'm no beginner - that the course is very useful to every beginner who is at least slightly ambitious and self-disciplined. There are a lot of lessons with good explanations and tons of exercices. Some of them require a rigorous, more or less difficult proof - or a reasonable guess, but most are fairly simple and provide ample opportunity for training. Often the questions are designed to make you look for a pattern or a way to reuse a previous result.
If you're closer to the frontier, the course may look less interesting, as it is being approached very late and possibly never reached. (Well, that's by design!) At the beginning, depending on your country, you may feel that you know most from school, but even during the earlier lectures a new teacher and perspective may give you exciting insights. During the last lectures I learnt some new stuff, which had not or only very superficially treated in my own numerical mathematics course (destined for physicists and engineers) at university (SVD, SVD & OLS estimators). Even if "foundations to frontiers" is slightly exagerated: it starts from zero and goes beyond the level of many university-level courses.
Thanks to Maggie Myers and Robert van de Geijn for offering this course!
Student
10/10 starsCompleted
5 years, 4 months ago
Really great introduction class to linear algebra. The time and effort that Robert and Maggie has placed in this course is first class. With their step by step approach folks without a strong math background can easily follow the material (which is not diluted by any means).
Would wholeheartedly recommend this course (this is from a person who has taken quite a few MOOCs).
- 1 review
- 1 completed
5 years, 4 months ago
Excellent course for anyone interested in Linear Algebra. Split into 12 weeks, the first 8 weeks introduce you gently, and the final 4 weeks pick the pace up with juicy topics such as basis transforms and eigenvectors. Each week has an enrichment section to put the new knowledge in context and offer some more advanced applications.
The lectures are very well put together, they are usually short and interspersed with homework questions to check you understood the video. This instant correction is really valuable. The exams have a practice exam first with video proofs which really help fill in gaps in knowledge.
Having taken about 10 MOOCs, I was really impressed with how much time Robert and Maggie spent answering questions and helping out on the forums.
Student
10/10 starsCompleted
5 years, 4 months ago
I took this course in the summer 2015 offering and I am very glad that I did. Even though I found the material challenging as I am in the declining years of my life, it was still worth the effort and it was very pleasing to be able to complete it.
The course is nicely structured and presented in a student friendly style with (short) videos and accompanying script. There is also a pdf of the material available as well which contains some material additional to the video material. It is advisable to watch/read both. The videos are short enough for the student to absorb without being overwhelmed with new information.
The material is organised into 12 sections. The first eight of these involved getting used to the FLAME notation and routines developed at UT Austin and putting them into practice in doing vector-vector operations, matrix-vector operations and matrix-matrix operations using Matlab. I found that having to program the o...
I took this course in the summer 2015 offering and I am very glad that I did. Even though I found the material challenging as I am in the declining years of my life, it was still worth the effort and it was very pleasing to be able to complete it.
The course is nicely structured and presented in a student friendly style with (short) videos and accompanying script. There is also a pdf of the material available as well which contains some material additional to the video material. It is advisable to watch/read both. The videos are short enough for the student to absorb without being overwhelmed with new information.
The material is organised into 12 sections. The first eight of these involved getting used to the FLAME notation and routines developed at UT Austin and putting them into practice in doing vector-vector operations, matrix-vector operations and matrix-matrix operations using Matlab. I found that having to program the operations deepened my understanding of them.
The remaining four sections were a step up in complexity, covering vector spaces, linear independence, bases, orthogonality, orthonormal bases, orthogonal projection, eigenvalues, eigenvectors and diagonalisation.
In each section there were online exercises to be done. Apart from programming exercises there were also calculation and multiple choice exercises. These latter were cleverly designed to elicit the properties of vectors, matrices and their operations.
In addition there were two midterm and one final exam. Preparing for these helped to consolidate the material. I found that I had to ask myself such questions as:
How do you go about LU factorisation?
How do you find a basis for the column space of a matrix? row space? null space?
How do you go about QR factorisation?
How do you handle an underdetermined system? an overdetermined system?
How do you project onto a column space?
How do you find eigenvalues? eigenvectors?
How do you diagonalise a matrix?
There was also a discussion forum where one could ask questions, make comments etc. The presenters Robert and Maggie were very prompt in replying. Their open-minded approach enhanced the feel good factor one got from this course.
Peter Schoofs
10/10 starsCompleted
5 years, 4 months ago
Great course! It starts very simple but step by step it develops to college level or under graduate math. By using Matlab (free time limited student license for use during the course) you get a good insight in computations with matrices. But…programming, even with an easy and intuitive programming language, can be time consuming. It's not necessary but I could not resist to do all the assignments. The 8 hours a week became about16 hours. Many extra material is offered for those who like practical applications and have an interest in math papers. For those with an interest in Quantum Physics: this course introduces matrix calculations, vector spaces, shifting to another orthogonal basis, and in the last week explores matrices with eigenvalues and eigenvectors. It's a nice course to start with linear algebra, or refresh your memory and extent your knowledge.
Student
10/10 starsCompleted
5 years, 4 months ago
The amount of work these professors and their teams put into creating this course is amazing. I have never taken and hardly even seen matrices in my past. It was a challenging and gratifying course. This course is one that really sets a high bar not only for MOOC courses, but offline courses as well. Massive amounts of slides where the professor walks you through the logical progression of the content. The two profs who put the course together really know how to get the relevant points across. If I were to make one suggestion, it would be to have a lengthier video earlier on in the course explaining the block matrix notation by itself, since so much of the course (and algorithms) depend on the student being able to connect the notation with what it would mean on matrices of different sizes. Well done to the Profs, everyone who put the course together, and University of Texas. This will serve as a foundational piece of my futu...
The amount of work these professors and their teams put into creating this course is amazing. I have never taken and hardly even seen matrices in my past. It was a challenging and gratifying course. This course is one that really sets a high bar not only for MOOC courses, but offline courses as well. Massive amounts of slides where the professor walks you through the logical progression of the content. The two profs who put the course together really know how to get the relevant points across. If I were to make one suggestion, it would be to have a lengthier video earlier on in the course explaining the block matrix notation by itself, since so much of the course (and algorithms) depend on the student being able to connect the notation with what it would mean on matrices of different sizes. Well done to the Profs, everyone who put the course together, and University of Texas. This will serve as a foundational piece of my future studies.
Student
10/10 starsCompleted
5 years, 4 months ago
Great Course densely packed with knowledge and well taught.
I took the course to get a better feeling for Linear Algebra, in anticipation of undertaking Quantum Mechanics courses. I have been well served by this course.
Maybe could benefit from more problems to get your teeth into, to practise and embed your newly acquired knowledge and skills.
One thing - the course has lots of really useful notes, but I'm verging on Dyslexic and can't learn by reading, I rely on videos - so more video demos please.
Student
10/10 starsCompleted
5 years, 4 months ago
I took this course as a refresher course. I had previously taken an Intro to Linear Algebra course that was strictly for math majors with no programming involved. It had been several years and I had only used linear algebra once since then. This course was perfect for me, because I also have a CompSci minor, so the focus on the algorithms for programming was very interesting to me. Note that the programming part is optional, you can do the course without doing the programming exercises (there is an option to skip those, and I had to skip them nearer the end due to time constraints). Also, this course simply helped me tie together the concepts much better than my first introduction to linear algebra, and covered topics that I don't even remember covering in my first class. I highly recommend to anyone interested in learning about linear algebra. Another note: although I had prior experience, I believe that the way the subject i...
I took this course as a refresher course. I had previously taken an Intro to Linear Algebra course that was strictly for math majors with no programming involved. It had been several years and I had only used linear algebra once since then. This course was perfect for me, because I also have a CompSci minor, so the focus on the algorithms for programming was very interesting to me. Note that the programming part is optional, you can do the course without doing the programming exercises (there is an option to skip those, and I had to skip them nearer the end due to time constraints). Also, this course simply helped me tie together the concepts much better than my first introduction to linear algebra, and covered topics that I don't even remember covering in my first class. I highly recommend to anyone interested in learning about linear algebra. Another note: although I had prior experience, I believe that the way the subject is presented would be quite good for a first time exposure to the subject. Excellent video lectures, materials, and solutions!
Student
6/10 starsDropped
5 years, 4 months ago
I'm afraid I couldn't get past the first week of the course. I had/have an interest in linear algebra, but my limited experience with computer programming was 35 years ago. I couldn't get past the complexity of installing and learning MATLAB as well as learning linear algebra. Those who dive in must really be prepared to spend the 8 hours/week.
Student
10/10 starsCompleted
5 years, 4 months ago
Marvellous course offered by UTA through edX.One find it easy going only with strong base of the subject.Concepts easily demonstrated in the videos and exercises can be completed well within the stipulated time.Content gets tough in Week 10,11,12 & requires more work. This is the lengthiest course I have done so far. Be calm and a lot of paper work is needed with usage of Matlab. The quality of the questions tickle your brain,while the detailed explanation ease the pressure.Takes a minimum of 10 weeks to complete,better if you upgrade to verified certificate early, I delayed and missed a certificate from UTA !!! I recommend to go ahead in Linear algebra those who have found glory and others can learn anytime easily.One can do 5-6 courses a year from the user friendly platform of edx and these require low bandwidth of data which soothes the long journey.
Many math courses are available where you build your future.
student
10/10 starsCompleted
5 years, 5 months ago
Time and time again, I have run into problems when applying linear algebra in a variety of other projects and courses. I have taken some linear algebra before, but it wasn't very practical. For example, about a year ago, I tried to write a program to solve a very large matrix with cramers rule. Needless to say, it didn't work out very well. If I had taken this course first, I could have easily made a program to solve that matrix in an efficient manner.
Overall, I feel this course taught the theory extremely well, while grounding that theory in practical programming. The last few chapters are a little rushed, but I suppose that's to be expected with an already-long course.
I highly recommend it to anyone wanting a very solid understanding of basic linear algebra, and exposure to many of the more advanced topics. This course exceeded my expectations.
Student
10/10 starsCompleted
5 years, 5 months ago
ROTFLAFFMAO
Absolutely The Best Math-Math Course I Have Ever taken. Absolutely The Best Learning Experience I Have Ever Had.
The best in all respects;
The best professors; clear, concise, knowledgeable, interesting, interested, concerned, leading edge contributors, expert presenters, easy to understand, provide a deeper view than in an ordinary course in this subject area. Prof v. d. Geijn’s participation level in the discussions board is amazing.
A great subject: of current topical interest to me, inherently interesting in and of itself, lends itself well to a MOOC format and schedule flexibility. The algorithmic emphasis is very valuable to me both here and beyond the course area – a new way to see the light in areas where it may have been neglected in our rush for answers.
A great course philosophy; to meet the needs of different students with differing objectives, needs, backgrounds.
A complete integrated l...
ROTFLAFFMAO
Absolutely The Best Math-Math Course I Have Ever taken. Absolutely The Best Learning Experience I Have Ever Had.
The best in all respects;
The best professors; clear, concise, knowledgeable, interesting, interested, concerned, leading edge contributors, expert presenters, easy to understand, provide a deeper view than in an ordinary course in this subject area. Prof v. d. Geijn’s participation level in the discussions board is amazing.
A great subject: of current topical interest to me, inherently interesting in and of itself, lends itself well to a MOOC format and schedule flexibility. The algorithmic emphasis is very valuable to me both here and beyond the course area – a new way to see the light in areas where it may have been neglected in our rush for answers.
A great course philosophy; to meet the needs of different students with differing objectives, needs, backgrounds.
A complete integrated learning system; outstanding lectures as well as complete background and supporting materials for mainstream/core and enrichment topics.
Interesting homework/exercises; which support the learning and are much more than rote application of formulas. Some are quite challenging (at least to me). The process of interleaving the lectures and the homework is very powerful.
The sequence of questions also brings insights.
Access to outstanding tools; MATLAB, FLAME, Spark, Timmy. Access to MATLAB alone is worth the price of the course. I wish I could justify a personal copy.
Student
10/10 starsTaking Now
5 years, 5 months ago
The course is very good at tackling simple matrix and vector operations. The many ways of slicing and dicing matrices and vectors aided my understanding of the material, and this perspective of slicing and dicing improved my grasp on the properties of matrices. Furthermore, coding on MATLAB also provided an excellent test of my understanding. The course offers many excellent exercises to show the properties of special matrices like upper, lower, diagonal, symmetric, and the identity matrix, and are supplemented by simple proofs in many notations like the FLAME notation. In the second part of the course, applications of matrix operations using slicing and dicing were given such as the QR factorization.
Even though LAFF is just an introductory course, I learned a lot even though I haven taken an introductory undergraduate course in linear algebra in the past. The enrichment activities and readings also made me appreciate linear alge...
The course is very good at tackling simple matrix and vector operations. The many ways of slicing and dicing matrices and vectors aided my understanding of the material, and this perspective of slicing and dicing improved my grasp on the properties of matrices. Furthermore, coding on MATLAB also provided an excellent test of my understanding. The course offers many excellent exercises to show the properties of special matrices like upper, lower, diagonal, symmetric, and the identity matrix, and are supplemented by simple proofs in many notations like the FLAME notation. In the second part of the course, applications of matrix operations using slicing and dicing were given such as the QR factorization.
Even though LAFF is just an introductory course, I learned a lot even though I haven taken an introductory undergraduate course in linear algebra in the past. The enrichment activities and readings also made me appreciate linear algebra techniques. For those who are totally new to linear algebra, it helps to go over the notes, exercises, and answers.
Student
10/10 starsCompleted
5 years, 5 months ago
This course is a very solid foundation for future work with linear systems. The pace of the course is well designed as well as how the various pieces fit together. In some mathematics course, there can be significant jumps along the way, but this course presents the material in a logical sequence without frustrating gaps in the explanations. It is obvious that a lot of thought and hard work went into creating this course, and I am very grateful to the instructors for all their hard work. As background, I had seen some of the material before taking the course but was self-taught. I worked for many years as an electrical engineer, so I have also seen some of the applications of linear algebra. This course filled in significant gaps in my understanding.
Student
10/10 starsCompleted
5 years, 8 months ago
Material presented clearly.
Their approach is different from other linear algebra courses by sharing MATLAB algorithms that eliminate most of the programming challenges working with matrices - dealing with element indices. They call the approach "slicing and dicing" and it helped work through a lot of problems.
Both instructors very responsive to questions posted on discussion board.
Both instructors speak clearly in their numerous videos.
Course notes are very clear and almost stand alone if you are a Reading Learner.
- 1 review
- 1 completed
5 years, 8 months ago
I found the homework AND grading system to be extremely helpful. there is immediate gratification on each step of the way and the instructors hand - hold you on the journey to learn linear algebra.
Student
10/10 starsCompleted
5 years, 8 months ago
The Perfect Course for a Computer Science Undergrad who is about to enter Graduate School after a break. Enabled me to gain a firm grasp on the concepts along with the modern day technology and algorithms used in the domain.
- 2 reviews
- 1 completed
5 years, 8 months ago
This course was so amazing, Excellent motivation, presentation and Exercises (to get insights and intuition). The staff was very active in the discussion forum. This course was like a dream, this was a FREE high quality education. Really guys I don't know how to thank you, you should so proud for what you offer
Student
10/10 starsCompleted
5 years, 8 months ago
Those who want to touch data science, machine learning, process, and/or text mining, computer graphics and recognition watch necessarily. Without orthogonal and singular decompositions, eigenvectors and eigenvalues, projections and normal equations at your fingertips there is nothing to do, ctrl+space will not help. Brilliant course by ,y mind.
Тем, кто хочет пощупать data science, machine learning, process and/or text mining, computer graphics & recognition смотреть обязательно. Без ортогональных и сингулярных разложений, собственных векторов и значений, проекций на подпространства и нормального уравнения на кончиках пальцев там делать нечего. Совсем нечего, ctrl+space не поможет.