Gilbert Strang might be good at linear algebra, but he's not very good in teaching. You are much better in that. No stuttering, no chaotic jumping over different topics and different parts of the blackboard, everything is clear and in proper order :) All with good, visual, geometrical intuitions and clear explanations.
@@MathTheBeautiful "But Grasshopper, someone must snatch the pebble," said Gilbert to Pavel. Agreed, Gilbert Strang is a legend. His OCW lectures were my introduction to linear algebra.
@@MathTheBeautiful Gilbert Strange is my No. 1 hero in algebra also . You are my No. 2 hero now ! Thanks for your teaching. Learn a lot from you.many thanks.:p
this lecture is more engaging than anything I've seen before, it really does make everything sound beautiful! Thank you for brightening my day and bringing a smile to my face!
Absolutely brilliant, so brilliant I went so far as to buy your book "Hello Again, Linear Algebra". Thanks for these wonderful videos and I wish you all the best for Lemma.
Ok this is like the best lecture. He actually motivates his explanations. Even me with my 2 braincells can figure out what he means. When he gives the length of the polynomial example, it really helped me to understand why I can't directly measure length. The intuition was very valuable. Thank you.
Please consider doing a video on weighted least squares to show how the projection is oblique under the standard inner product, but orthogonal under the 'right' inner product.
Trully the best way to approach linear algebra of vector spaces. Not to teach how to solve it, but to actually give a deeper understanding of WHY we are doing it. I am a structural engineer and had to learn it the hard way, on my own because in college we only learned how to do it. :) Great vid!
. . . . . . . ** . . . . . . . . ** What is☝☝☝THIS or ☝☝☝ THIS?? I often see this notation in mathematical writings. To me, they both look like inner products, but with THREE inputs. How do you go about evaluating these? What is the proper interpretation of this notation?
See ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Psb1Zuxo7gs.htmlsi=3E1stR16in-S0gyD&t=250 for one possible analogy. Another analogy is that it's the inner product of the vectors 𝜓 and (𝚽𝜙)
After reading my current textbook and didn't get a lot, was surfing youtube for an explanation why Inner product is needed and it seems that this is the vide i was looking for. I believe the worth trying resource for sure. Thanks!
You are right, we have always been trained to assume "inner product" as just "length". Inner products, as you mention are far more fundamental than attributes such as lengths, angles (for geometric vectors). The nature, perhaps, must be using inner products to compare two objects (A, B) with respect to a chosen set of attributes. In the case of geometric vectors, objects A and B are vectors, and an attribute that we "chose" to do the comparison is length. If we compare two surfaces A and B, the attribute perhaps can be chosen as area. If the surfaces are identical but differ only in roughness, then choosing just area wouldn't suffice to tell whether A and B are identical or not. Then we have to compare both area and roughness. If two surfaces A and B have the same area and also roughness, but differ only in color, then we need to include color as an attribute for comparison.
What makes it so obvious that length is the right measure of how accurate the (semi) solution is in the case of your rectangular matrix multiplication? Why not minimize the sum of the errors? I know its a more convenient calculation and it uses the power of matrices, but is that the only reason?
You're exactly right. There isn't one best preset measure to be minimized. The choice of measure should depend on the particular problem you're trying to solve. Whatever measure you choose would be called "length". Some lengths come from inner products, some (like the sum of |errors|) don't. The ones that come form inner products have some advantages. Other measures, like the one you're suggesting, have other advantages.
