Linear Algebra - Foundations to Frontiers
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Course Details
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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,
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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!
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Student
10/10 starsCompleted
4 years, 8 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
4 years, 8 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
4 years, 8 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
4 years, 8 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
4 years, 8 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
4 years, 8 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
4 years, 8 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
4 years, 9 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
4 years, 9 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
4 years, 9 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
4 years, 9 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 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 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.
- 2 reviews
- 1 completed
5 years 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 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.
Student
10/10 starsCompleted
5 years ago
Everything about this course is amazing! I don't usually write reviews that are so unilaterally good but this class really fits the bill. The instructors have put so much effort into the course and it really shows. I'm so glad that I took this class.
Student
10/10 starsCompleted
5 years ago
may be it will be better to divide this course in two parts... sometimes it is hard to follow a mooc with 15 weeks of length because of other duties... i prefer to have moocs with 7-8 weeks. anyway it was a wise think to do: open all the deadlines till the end of the course.
Student
10/10 starsTaking Now
5 years ago
In short, I now understand how linear algebra works. I see the matrix or an equation with matrices and understand what kind of manipulation it does. Previously it was a black box for me.The platform, materials and forum are great. Instructors reply to forum questions within hours or a day and their replies are always helpful.
Student
10/10 starsCompleted
5 years ago
It was a great time spending these 12 weeks with LAFF staff!! I had linear algebra courses during my university time so many years ago but this LAFF course is really digging into the foundations and the reasons of the topic. I took several MOOC also in other platforms; the LAFF course is definitely one of the best I had. Once again I have to thank the staff for dedication to the course. I am looking forward to take one more course in an advanced topic about linear algebra with LAFF staff !!! Thanks Robert, thanks Maggie!!
Student
10/10 starsTaking Now
5 years ago
This class is the best math class I've taken, on or offline. It is obvious how much care and craft went into its planning and execution. The organization is also exceptionally clear and easy to navigate compared to most MOOCs.
As a former English major who has been away from math for about 15 years, I found this class challenging to say the least! But I also found that there were enough resources to overcome most trouble spots.
- 1 review
- 1 completed
5 years ago
I have done several MOOCs so far and, by my experience, they fall into 4 categories:
1 - Those that does not deserve any mention (poor material, poor teaching strategy, poor teacher/student interaction...)
2 - Those that have a motivated and participant instructor that unfortunately fails to provide a good learning environment.
3- Those that have a very well crafted material that, by itself, is able to deliver good results to engaged students, even with the lack direct interaction between instructors and students.
4 - Few that are able to provide not only a very well made material as also have instructors that are really engaged in the course all the time.
This course stands into the latter.
Professors Robert and Maggie not only presents the course masterfully as also interact and respond every single question posted in the forums.
Also, you will find most of the exercises and demonstrations very well written and many of those...
I have done several MOOCs so far and, by my experience, they fall into 4 categories:
1 - Those that does not deserve any mention (poor material, poor teaching strategy, poor teacher/student interaction...)
2 - Those that have a motivated and participant instructor that unfortunately fails to provide a good learning environment.
3- Those that have a very well crafted material that, by itself, is able to deliver good results to engaged students, even with the lack direct interaction between instructors and students.
4 - Few that are able to provide not only a very well made material as also have instructors that are really engaged in the course all the time.
This course stands into the latter.
Professors Robert and Maggie not only presents the course masterfully as also interact and respond every single question posted in the forums.
Also, you will find most of the exercises and demonstrations very well written and many of those with videos in which the instructors themselves guide you step-by-step.
I took this course as a refresher, since I had a previous undergraduate course in Linear Algebra but, not only I was able to review many of the topics I wanted, as also I was presented to some techniques (Flame notation, slicing and dicing...) that really improved my knowledge.
There is only one negative aspect of the course in my opinion, and that is their choice to use matlab in the programming assignments instead of python from the last offering.
However, even with that, the material is well documented and there are examples in python as well (but I really hope they change back to python in the next editions)
Anyway, it was an amazing course and I really recommend it to anyone willing to learn or review the topics of a undergraduate course in Linear Algebra.
Student
10/10 starsCompleted
5 years ago
I learned Linear Algebra before in school but didn't understand what I was doing. I just recited the calculations (Linear equations, LU, SVD, eigenvalues) and wrote it on the exam papers. It didn't left much or made some changes in my mind after the exam
After I graduated from school and found LA is really really really important (Can't express the importance, it's just the language of some subjects).
Therefore I took this course, the course well covers the foundations of Linear Algebra which you can find out from typical textbook (except Det) and most importantly, it will teach you how to think and make sense of concepts such as matrix, vector, factorization, eigenvalue, SVD, LU, QR factorization.
This course builds my solid foundation and I can go further based on these 12 weeks. The course is easy to follow, self-paced [no deadline except final which is excellent policy for worker]
As Robert wrote, A typical undergrad...
I learned Linear Algebra before in school but didn't understand what I was doing. I just recited the calculations (Linear equations, LU, SVD, eigenvalues) and wrote it on the exam papers. It didn't left much or made some changes in my mind after the exam
After I graduated from school and found LA is really really really important (Can't express the importance, it's just the language of some subjects).
Therefore I took this course, the course well covers the foundations of Linear Algebra which you can find out from typical textbook (except Det) and most importantly, it will teach you how to think and make sense of concepts such as matrix, vector, factorization, eigenvalue, SVD, LU, QR factorization.
This course builds my solid foundation and I can go further based on these 12 weeks. The course is easy to follow, self-paced [no deadline except final which is excellent policy for worker]
As Robert wrote, A typical undergraduate linear algebra course offering includes a sequence of three classes:
1 Introductory Linear Algebra. [This course]
2 Applied Linear Algebra.
3 Numerical Linear Algebra.
I am really looking forward to the latter two!!!
- 2 reviews
- 2 completed
5 years ago
This is an exceptionally good foundation course to linear algebra. It will give you a strong intuition of how matrices work and why they behave the way they do. I learned more by taking this course than I did during university.
Student
10/10 starsCompleted
5 years ago
Preporučam ovaj tečaj svakome tko nastavlja bilokakvo školovanje dalje od srednje škole, kako zbog sadržaja, tako i zbog naglašeno pedagoškog pristupa. Srednjoškolcima također, budući da započinje osnovama linearne algebre, i onda dalje u nešto apstraktnije vode.
Student
10/10 starsCompleted
5 years ago
The Linear Algebra Foundations to Frontiers (LAFF) course taught by Dr. Robert van de Geijn and Dr. Maggie Myers is outstanding. I have an Honours Bachelor of Science degree (Biology/Forensics) but did not really have any linear algebra exposure beyond high school. I have recently developed an interest in statistics and computer programming and thought it would be useful to learn more about linear algebra, so I signed up. I’ll admit that I was intimidated at first, but after only a short time, I realized that this course was made for people like me with very little prior experience in this field; the concepts are carefully presented so that each builds upon the previous work with plenty of opportunities to reinforce the material through homework assignments, two midterm exams and a final exam. It is both challenging and yet, surprisingly intuitive, which I believe is because of how well the material is presented and also, the arr...
The Linear Algebra Foundations to Frontiers (LAFF) course taught by Dr. Robert van de Geijn and Dr. Maggie Myers is outstanding. I have an Honours Bachelor of Science degree (Biology/Forensics) but did not really have any linear algebra exposure beyond high school. I have recently developed an interest in statistics and computer programming and thought it would be useful to learn more about linear algebra, so I signed up. I’ll admit that I was intimidated at first, but after only a short time, I realized that this course was made for people like me with very little prior experience in this field; the concepts are carefully presented so that each builds upon the previous work with plenty of opportunities to reinforce the material through homework assignments, two midterm exams and a final exam. It is both challenging and yet, surprisingly intuitive, which I believe is because of how well the material is presented and also, the array of tools that are used to help visualize the concepts (Spark, PictureFLAME, Timmy, MATLAB). The emphasis is on truly learning and understanding the material rather than just achieving a passing grade; the instructors offer unlimited tries for every question and comprehensive answer keys are provided with each question so the student has the ability to try solving the problems as many times as necessary without penalty and is never left wondering how to find the answer. The instructors are active on the discussion board and always willing to answer questions. It was an amazing opportunity to be able to complete this course for free, and I would highly recommend it to anyone with an interest in the STEM-related fields (Science, Technology, Engineering, and Mathematics), but especially to women, since we are underrepresented in all of those fields - try it out! If you complete every assignment and watch every video (which I recommend doing), the estimated 8 hours/week time commitment was accurate, but it IS absolutely worth your time to LAFF with them.
-Kathy D.-
Student
10/10 starsTaking Now
4 years, 12 months ago
Excellent. The only limitation was my own time and ability. While I did not complete the course, I will keep trying and now with a better foundation. The amount of effort in preparing this material and presenting it, is nothing short of remarkable. Thanks!
Student
10/10 starsCompleted
4 years, 12 months ago
I took this course because I had tried to take the Stanford Machine Learning course but discovered that I did not know/remember enough linear algebra. (Actually some of the subjects did not exist when I first took linear algebra in college.) The linear algebra review included in the machine learning course was not adequate. This course, however, more than adequately prepared me to take the machine learning course. In fact, several of the subjects included in this course were described as advance linear algebra subjects by Dr. Ng in the machine learning course. In addition, I now feel quite comfortable with matrix notation and even Matlab vectorization.
Previously I had tried to relearn linear algebra with Dr. Strang's book and lectures, but the lack of "homework" problems with good feedback was not adequate.
This course covers the material well and has good online homework with excellent backup material.
Thank you
Student
10/10 starsCompleted
5 years ago
I took this course thinking in making some kind of refresh to then make a better study of Digital Signal Procesing. But I was surprised of the whole bunch of things that I didn't know about the foundation of Linear Algebra. There is a lot to learn in this course, and a lot of work of course! not to be taken lightly.