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Thanks. You might enjoy some companion resources: jeff-anderson-wru2.squarespace.com/s/Anderson_M2B_Lesson_17_Notes_v20190403.pdf jeff-anderson-wru2.squarespace.com/s/Anderson_M2B_Lesson_18_Notes_v20190403.pdf For more stuff like this, check out: www.appliedlinearalgebra.com/blog/for-students/welcome-to-math-2b I'm working on a textbook that you can read more about here: www.appliedlinearalgebra.com/ Thanks for your attention and cheers to your learning!

Working on it. My next releases will be to fully replace this with a much better version.. Here is a draft of a textbook section on Permutations: www.appliedlinearalgebra.com/s/Anderson_Applied_Linear_Algebra_Introduction_to_Permutations_Draft_v20240330.pdf I'm currently producing videos to go along with that work. I do have other resources to help with this that are currently available: jeff-anderson-wru2.squarespace.com/s/Anderson_M2B_Lesson_16_Notes_v20190403.pdf jeff-anderson-wru2.squarespace.com/s/Anderson_M2B_Lesson_16_Permutations_Handout_v20190403.pdf jeff-anderson-wru2.squarespace.com/s/Anderson_M2B_Lesson_16_Permutation_Worksheet.pdf Thanks for the comment and your attention, Stay tuned for much richer content in the future.

Thanks. If you like this, you might like these too: www.appliedlinearalgebra.com/blog/for-teachers/linear-algebra-laboratory-exercises/make-eigenvalues-resonate www.appliedlinearalgebra.com/linear-algebra-laboratory-exercises/linear-algebraic-nodal-analysis All of this is part of much larger project: www.appliedlinearalgebra.com/ Cheers to your learning and thanks for your attention!

Hello Nick. Thanks for the comment. I support my students in using digital planning tools. But for a bunch of reasons, I like to stay in analog for this type of work. The book "Bullet Journal Method" by Ryder Carroll has some cool ideas about why analog notebooks can provide lots of benefits compared with digital planning tools. I'd love to chat through this during class, if you wanted to talk more about it. Thanks for the comment and I look forward to many more meaningful discussions with you.

Sorry for the delay: I took a little break from work. Great question! To learn more about what that dot means, you might enjoy the following video: ruclips.net/video/ngBW5dnW65c/видео.htmlfeature=shared For a much more comprehensive list of my Linear Algebra videos (which includes many that discuss outer products and rank one updates), check out the following list: www.appliedlinearalgebra.com/s/Math_2B_Playlist_Video_Titles_and_Video_Lengths_Draft_v20220418.pdf Your question about where we use rank one updates is a good one. Without thinking too hard, I can give the following list: 1. In the definition of elementary matrices to do Gaussian elimination 2. in the definition of householder transformations ( en.wikipedia.org/wiki/Householder_transformation ) for numerical algorithms (this is a very popular tool in numerical using linear algebra) 3. In the outer product definition of matrix-matrix multiplication 4. In optimization algorithms like BFGS algorithm ( en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm ). 5. In many algorithms for data science. To learn more about that, you might like the following book: math.mit.edu/~gs/learningfromdata/ I would draw your attention to a section on this exact topic: math.mit.edu/~gs/learningfromdata/dsla1-2.pdf 6. The sherman woodbury formula for inverse of a rank one update: en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula I am in the middle of some other work right now so I will limit my response to what I have typed above. But, given a little more time, I know I generate many more examples of where rank one updates can be used (and are used in practice). Rank one updates are a fundamental tool in linear algebra, kind of like the power operation in scalar arithmetic. Does this answer your question? Thanks for your attention and cheers to your learning.

Thanks for this kind feedback. Sorry for the delay: I took a little break from work! If you like this video, I think you'll love my next releases. I'm planning to create a much more detailed version of this video including good visual effects and a sequence of textbook sections that expand on this work. For a sample of the type of work, check out the following video: ruclips.net/video/25wiDY8OIoQ/видео.htmlfeature=shared That is a rough draft of the first few minutes of my next release. That video will be the first in what I hope will be a series of three videos that together will be an Introduction to Permutation. Here is my most recent draft of the correspondin textbook section: www.appliedlinearalgebra.com/s/Anderson_Applied_Linear_Algebra_Introduction_to_Permutations_Draft_v20240330.pdf This introduces a bunch of good details on what permutations are, how they work, and how to calculate the sign (which is used in the definition of the determinant). After this set of videos, I will move on to two more series. One will be one the geometry of elementary transformations. The other will be on how to discover the determinant function. In other words, it will be designed to improve this video that you are commenting on. I hope to release this content in the coming months and years. My larger goal is to write an entire textbook that I give away for free. For more about this, check out my project homepage: www.appliedlinearalgebra.com/ Thanks again for your attention and cheers to your learning.

You're most welcome. Just wait. I'm so excited for my next phase of my RUclips career. I plan to write a full textbook, available for free, where each section includes a RUclips series with really good graphics. Here is a sample of the textbook section for my next releases: jeff-anderson-wru2.squarespace.com/s/Anderson_Applied_Linear_Algebra_Introduction_to_Permutations_20240309.pdf Interesting enough, the content is related to this video you've commented on since I highlight more details about permutations. Here is a sample for the corresponding video for the first part of that textbook section. ruclips.net/video/25wiDY8OIoQ/видео.htmlfeature=shared To learn more about the textbook project, check out the following link: www.appliedlinearalgebra.com/ Thanks for your attention and cheers to your learning.

Thanks. I cannot wait for my next releases on RUclips. I plan to produce much higher quality graphics and also include a full drafted textbook section with each release. For a sneak peak of the work, check out one of my more recent uploads: ruclips.net/video/25wiDY8OIoQ/видео.html That was a test of new video production skills I'm building. That video is one of a series of three to translate content about permutations into high-quality RUclips videos. Below is a link to the draft content for this upcoming video series: jeff-anderson-wru2.squarespace.com/s/Anderson_Applied_Linear_Algebra_Introduction_to_Permutations_20240309.pdf This is part of a much larger textbook project. In addition to being freely available online, this textbook will also feature high-quality RUclips videos (like 3blue1brown). Here is my current draft of the table of contents for my work: jeff-anderson-wru2.squarespace.com/s/Jeff_Andersons_Applied_Linear_Algebra_Fundamentals_Table_of_Contents_draft_v20240227.pdf Here is a brief description of the larger project: jeffandersonmath.wordpress.com/2023/03/08/jeff-andersons-applied-linear-algebra-fundamentals-textbook-project/ I am working to produce full solutions to the corresponding beginner-level questions so that interested viewers/readers can check their work. Thanks for the kind note and cheers to your learning. PS. If you like learning about linear algebra, you might enjoy some of the resources I highlight int eh following blog post: jeffandersonmath.wordpress.com/2022/05/11/more-linear-algebra-please/

Me too! This is just so cool to see.

Right? It's just amazing to see what happens when students work hard and are set free to explore and learn. Sending you love today and I hope you are thriving!

Thanks for the kind comment. If you liked this video, perhaps you'd like some of my other linear algebra videos: www.appliedlinearalgebra.com/s/Math_2B_Playlist_Video_Titles_and_Video_Lengths_Draft_v20220418.pdf And, if you're interested in learning to code this stuff in MATLAB, here are some introductions to the basics: www.appliedlinearalgebra.com/s/Anderson_MATLAB_Master_Video_Playlist_Draft4_2023_01_08.pdf You might also enjoy the laboratory exercises that I've made to support student learning on applied projects: www.appliedlinearalgebra.com/blog/for-teachers/linear-algebra-laboratory-exercises The goal of all of this is to help support you in developing meaningful expertise that helps you build a career you love. In the coming months and years, I'll be release some very fun projects. For a small sample, check out: jeffandersonmath.wordpress.com/2024/01/31/jam-project-updates-1-31-2024/ Thanks for your attention and cheers to your learning. You are brilliant, strong, and powerful. I know you can learn math at the highest and deepest levels.

Thanks for the comment. I'm not quite sure what 10000 means? I do hope this video is supporting you in learning new skills. If you like this video, you might like some of my other MATLAB videos. You can find my current list (as of the day I'm writing this comment) in the pdf below: www.appliedlinearalgebra.com/s/Anderson_MATLAB_Master_Video_Playlist_Draft4_2023_01_08.pdf You might also enjoy my MATLAB course homepage: www.appliedlinearalgebra.com/blog/for-students/welcome-to-engr-11 On that page, I share other resources to support fun learning in scientific computation. Thanks for your attention and cheers to your learning.

Thank you! I've been thinking of you this week: I walked past that pull up bars on campus and wanted to do some reps in your honor. I also have three students who say they want to code up a neural network so I've been talking about you. I hope you're thriving. In terms of this video, I'm so excited for my next releases in the coming months/years. I am learning a lot right now and am so excited to up my game for my students! I'm on the hunt for 1 million subscribers so I can fund the creation of a fantastic textbook in linear algebra that is available for no required paywall. Thanks for the note and happy Friday.

Thanks for your feedback. I'm wondering which former student this is? I'm very excited for the next stage of development on this! I can't wait to be able to provide excellent textbook content than includes high quality videos with no required paywall! For more about this, check out the link below: jeffandersonmath.wordpress.com/2023/03/08/jeff-andersons-applied-linear-algebra-fundamentals-textbook-project/ I wish you continued success in your education and beyond. Thanks for the note!

@@JeffAndersonMath Thanks for the reply. My name is Pranav, I was in your Math 2B class before going to Cal. I too have been running an educational-type channel (this one!). Glad that you are taking this initiative to make more free, high-quality content. Good luck, and thank you!

@@iKnowOfficialYT Hello Pranav. I remember your name and I'm having a hard time remembering your face at the moment. Cheers to going to cal and congratulations on your RUclips channel. About 6 months ago I decided to take RUclips seriously as a business so that I can generate income to pay for the creation of my mathematical content. I'm having a blast with this work and I am working to build a channel with 1+ million subscribers so that I can give away excellent mathematical material for no required paywall. This video is a teaser to the first project I am releasing with my new approach. And I can't wait to keep this going. I believe students deserve excellent educational materials and I love the idea of making these available for free without needing to compete for grant money that provides short-term funding for long-term projects. This is part of my linear algebra textbook project, which you can learn more about here: jeffandersonmath.wordpress.com/2023/03/08/jeff-andersons-applied-linear-algebra-fundamentals-textbook-project/ To be honest, the linear algebra textbook is a support resources for high-quality applied math modeling laboratory experiences that help students learn how to apply linear algebra to solve real-world problems related to their majors. For more about my effort to create these types of laboratory experiences, take a look at the manuscript for a paper I drafted (with a previous student) and submitted for publication which you can find at the follow URL: www.appliedlinearalgebra.com/s/PRIMUS_REVISION_Linear_Algebraic_Nodal_Analysis_Anderson_Draft_20231030.pdf Cheers to your learning, congratulations on your work at CAL + your RUclips channel, and thanks for your kind words. Stay tuned in the coming months for a lot more fun!

I've had so much fun learning how to do this... And this is the start. I've been working on the video for about 2 months. The textbook section I wrote over the summer. Last spring, I wrote up my plan on how I will use the income I generate. I can't wait to see your channel and your content! Anything I can do, you can do better!

Great catch. I think this was a copy and paste error. To make that definition, I copied my template from the column-partition version and think I missed the switch from k to i. However, later in the video, it seems like I use the proper form of this definition to run through the examples I provide. Thanks for pointing this out, for your attention and for your keen eye. Cheers to your learning!

Hmm? What does this mean? Perhaps these are the values of the coefficients? If you liked this video, perhaps you'll like some of my other linear algebra videos. To see my current list, please check out my Linear Algebra course homepage: www.appliedlinearalgebra.com/blog/for-students/welcome-to-math-2b Then, scroll down to the "Anderson’s Master List of Applied Linear Algebra Videos (.pdf)" link and click. Alternatively, you can check out any of my "Lesson N Playlist" links on that page, where N = 0, 1, 2, ..., 18 etc. Thanks for your interest and your attention. And cheers to your learning!

Nevermind. The next video clears it. I did not recall that we used one entry of a column in matrix A as pivot position to zero out a particular entry in the same column

I love your question. Thank you for engaging like this. The way I define pivot columns, at least in my mind, are the linearly independent columns of the matrix when looking from left to right. If we start at column 1, let's ask ourselves: is column 1 nonzero. If column 1 is nonzero, then it is a pivot column. Then, we go to column 2 and ask the question: is column 2 a scalar multiple of column 1. If column 2 is not a scalar multiple of column 1, then it is a pivot column. Then we go to column 3 and ask ourselves: is column 3 a linear combination of columns 1 and 2. If column 3 is linearly dependent on columns 1 and 2, then it is a nonpivot column. On the other hand, if column 3 is linearly independent from columns 1 and 2, then column 3 is a pivot column. We continue in this way until we classified every column as either a pivot column or a nonpivot column. Because it's not always easy to be able to ascertain this information by looking at the original columns of the matrix, we can use elementary row operations (by multiplying the original matrix on the left by our elementary matrices: shear, dilation, and transposition matrices) to transform the original matrix A into U = rref(A). In that case, we can identify the pivot columns as corresponding to the columns of U with 1 in the leading entries with all other entries being zero. That is a convenient way to identify which columns are pivot columns and which are nonpivot columns. But, in my mind, the real point of the distinction is to help us figure out which columns of the original matrix A are linearly independent and which are redundant on previous columns. What comes up for you as you read my response? Does that answer your question?

Thank you for your kind words and feedback. If you like this video, you might enjoy some of the other ones I have available for linear algebra: www.appliedlinearalgebra.com/s/Math_2B_Playlist_Video_Titles_and_Video_Lengths_Draft_v20220418.pdf or for learning how to code in MATLAB: www.appliedlinearalgebra.com/s/Anderson_MATLAB_Master_Video_Playlist_Draft4_2023_01_08.pdf You can also find some good learning resources for my linear algebra class on my course homepage: www.appliedlinearalgebra.com/blog/for-students/welcome-to-math-2b In the coming months and years, I'll be producing videos that come along with well-written textbook sections to support deeper learning from the video. Keep your eyes posted for more on that. Thanks again for your attention and cheers to your learning.

Wow! I love this question. Just to make sure I'm understanding, you mean the transpose of the output matrix, correct? In my videos, notes, and resources, I do define important differences between the term transpose and the term transposition. In this case, I do follow standard conventions in our field. You can learn a bit more about the differences between these terms on wikipedia: en.wikipedia.org/wiki/Transpose en.wikipedia.org/wiki/Cyclic_permutation#Transpositions Assuming that your question is about the connection between the transpose operator and the ability to switch the order of factors in a matrix-matrix product: YES! There are deep connections between switch the order of the argument and the transpose operator. The mathematical formula that comes to mind at this moment is: (A* B)^T = B^T * A^T In nerdy language, I would write: when we take the transpose of a matrix-matrix product, that is equivalent to taking the product of the transposes of the factors in reverse order. It's a very fun fact in linear algebra that taking linear combinations of column vectors is different than taking linear combinations of row vectors. The transpose gives us the ability to discuss how these operations are related to each other. I have so much more to say and I'm overjoyed for you that you're starting to pull on this thread. I'll leave you with this: you're starting to set the foundations for understanding the fundamental theorem of linear algebra: mathworld.wolfram.com/FundamentalTheoremofLinearAlgebra.html Lessons 0 - 18 in our class set a very strong foundations for understanding that theorem. Section 2.5 fromthe book Applied Linear Algebra by Olver and Shakiban also has a great discussion of that theorem: warin.ca/ressources/books/2018_Book_AppliedLinearAlgebra.pdf This is related to work that Gilbert Strang has done to discuss the four fundamental matrix subspaces and sets the stage for great algorithms to solve least squares problems: web.mit.edu/18.06/www/Essays/newpaper_ver3.pdf Ok. I'll stop here. What comes up for you as you read this response? Cheers to your learning and thanks for this great question.

@@JeffAndersonMath Just found this response a month later, and going back through this was a fun little side quest. I remember writing this comment as I was just trying to wrap my head around MMM, and seeing several different approaches to it and realizing there are multiple ways to do it helped concrete my understanding. Quickly reviewing this video I don't see you mentioning any transpose operations, so I realize that connection was something explicit I made during my study. I suppose I felt good enough about the observation to leave a comment about it, and your response now furthered that connection even a month later haha. This class is the gift that keeps on giving, I now have a more detailed reminder of how transpositions are individual subset operations of a transpose, or any permutation. Reading about the FTOLA if I boil it down the right way sounds intuitive, but I recognize it goes way deeper and I understand maybe 50% of what's going on with that article. I hope to keep bumping into this material in my free time and continue my learning much beyond our ten week window

Kind of. This really gets into how the cross product is constructed. It turns out the cross product encodes information about the "volume" of four different parallelograms that we can create by looking at two vectors in R3. To learn more about this, you might enjoy my Math 1C, Lesson 4 resources that you can find at the link below: www.appliedlinearalgebra.com/blog/for-students/welcome-to-math-1c The videos and notes are a bit long but they help you build very deep intuition for the connection between cross products, areas of parallelograms, and 2-by-2 determinants.

Please do feel free to ask this question in class, btw. I love how you track and ask your questions. I wish all my students worked like this.

Thank you for your attention and positive feedback.If you like this video, perhaps you might enjoy some of my other videos also. You can find the current draft of my linear algebra video playlists here: www.appliedlinearalgebra.com/s/Math_2B_Playlist_Video_Titles_and_Video_Lengths_Draft_v20220418.pdf You can find my applied linear algebra course homepage here: www.appliedlinearalgebra.com/blog/for-students/welcome-to-math-2b That homepage includes links to a draft textbook manuscript and many other fun resources for learners. In the coming months and years, I will be releasing some very special linear algebra content that includes videos that have much higher production value (something like 3blue1brown) along with links to a textbook manuscript where viewers can learn the material from the video from a free online textbook. Keep your eyes pealed for more on this. Thanks again for your attention and cheers to your learning.