1. Markus Reinert Author

    Not quite what I expected from the title, but he is great! Watching the video(s) is very entertaining and informative. Thank you!

  2. johnmartin1024 Author

    Dearest Gil, It is beautiful and wonderful to watch you work. You are truly a gifted expositor in Mathematics. You bring richness to students of all ages. John M.

  3. Victor Novikov Author

    for everyone looking for even bigger and easier to digest picture https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

  4. semih oguzcan Author

    Every lecture should have a "the big picture" part as this one has. Unfortunately in many cases the lectures are done just to do lectures, not to see the "big picture".

  5. Marty Lacerda Author

    What did he mean by a perpendicular line is an object in ONE dimensional space? Wouldn't it be two dimensional? Height and width?

  6. Zach Grant Author

    It's kind of amazing that there are so many professors out there that don't succeed in explaining this to people, but someone that watched this video could take the information learned and explain it to a fellow student successfully in a short amount of time. I will admit it is possible that youtube videos such as these are doing no more than filling in gaps of professors, instead of informing the student 100 percent more than they were before they watched it.

  7. Double You, Bee 11 Author

    Watch the TA video on Ax=0 to understand the perpendicular concept. https://youtu.be/LttE1vDVrm0?t=5m29s
    Very helpful diagram at 5:29. Before seeing that, I didn't really understand the concept of null space and plane.

  8. mathIsART Author

    So the Big Picture (in technical language) seems to be: 1) the row space is the image of the matrix, 2) the null space is the kernel of the matrix, 3) the column space is the dual of the row space, 4) the left-null space is the dual of the null space

    Thank you, Gilbert Strang <3

  9. Omega Point Singularity Author

    3:50 whats happening, "one of the third one of the second" and so on, how can you make a 3d vector like that.. how can you multiply a 3d vector to a 2d plane like that?

  10. Simon Demarque Author

    for a foreigner, if I understood well, the 'left' means, remaining ? right or false ?
    but 'remaining' takes much more place to write on the blackboard, that might be the reason of using the vocable 'left'

  11. Phisit Author

    I was studying for my molecular genetic exam and some how ended up here. It started off well then I was lost when he started talking about " Plain ", first thing pops up in my head is Boeing aircraft plane. This comment is pointless but thank-you for reading.

  12. Wade He Author

    When you go with a 1.5 OR 2 speed, dear Prof. Gil is quick and vigorous like a young man! It's the fascinating part of online learning.

  13. Paul Mina Storm Author

    so various combinations of A are basically vector points that essentially make up two linear lines (because 2 vectors are given here) on a plane. But deriving the null vector N(A) (the perpendicular line to the A space plane) gives you every other possible vector that can sit on the plane. Elegant! So increasing the vector dimensions still allows this method to be used to calculate solutions for multidimensional spaces and I guess N(A) may have multiple solutions that satisfy N(A) thus the solution space may be a line (only one null space vector is possible) or a higher dimensions (more than one N(A) solution). Of course everything has to be calculated relative to the zero origin (because a line or plane does have infinite perpendicular lines/solutions otherwise). So the whole point of this is to quickly be able to determine if a solution (ie multidimensional vector) is in a particular space or not. It seems interesting for applied problems (eg AI, perhaps graphics related) but I feel LA may reach its limits in certain applications simply because it is fundamentally 'linear' (vector based) after all.

  14. Mathew Schau Author

    Is there a reason why the null space is perpendicular to the row space? 5:23 The explanation only proves that it is perpendicular but not why it would be in the first place.

  15. eswyatt Author

    Implies that although, in an nXm matrix, n may not equal m, the number of independent rows always equals the number of independent columns.

  16. Yizhong Sha Author

    The left null space is explained using the wrong diagram, it would be much clearer if the professor draw a new diagram for the transposed matrix instead of re-using the column space diagram for the regular matrix


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