“Universal Property” is the fundamental LAW of modern math, appearing in the Category Theory, Hopf Algebra, etc.
Prof Penn brings this abstract concept down to undergrad maths using the 2 examples : Derivative (f’), Determinant (det)
ie without knowing the end-result (f’ or det) , by just following their Universal Properties , we can achieve the same.
[Video 5:50 mins] Excellent technique in 2 Product sum series: red box to yellow box (swap inner & outer sums, change index), and the diagram visual trick.
Category Theory explained in 1 min via concrete example:
By definition Category (aka “Big Set”):
Objects with Morphism (aka Functor, arrow, relation)
When we learn abstract algebra, we have never been clear to check
“Well Define” definition ?
“Well Defined” Function 的 含义：
不会“因人而异“
Regardless of what (elements : x or y)， the definition still valid.
See how Michael’s proof 👍
Prove :
“2 Cosets Equal” Lemma: (Red box => Yellow box) … all 5 lemmas proved with “Set Inclusion Technique”
Penn’s teaching is excellent, he can link from Sec School Elementary math (Trigo, Analytic Geometry) to Higher Math (Inner Product, Vector Space,Bi-Linear form, Orthogonal Matrices).
《从高数看初等数学》 was the idea of Felix Klein, the German mathematician cum Math educator who built Gottingen University.
Klein introduced “Function” into all Secondary Math curriculum from Sec 3.
Let z = x + i.y,
find |cos z|² = ?
The scary Group Representation (群表示论) is best explained by the friendly online Prof Michael Penn.
French Classe Préparatoire teaches Group and Vector space as two distinct topics, without the “Group Representation” which actually links both.
北大 /清华 丘维声 教授 的 《群表示论》 从群 Group 的 “零基础” 教起 。
https://v.ixigua.com/Y7tYgR1/
Penn showed his powerful technique in double sums: change 2 indices (m,n) by combining with new index (k=m+n).
Continuity vs Uniform Continuity? the former depends on point ‘a’, or the later independent of ‘a’.
Penn is a great Math Prof from an unknown USA college, but his lectures so easy to understand, even the Math problems involve complicated tricks & techniques.
Part 2: Theorem for “Not Uniformly Continuous”:
Beautiful Proof using the French Bézout Theorem (of GCD & Co-Prime).
Useful Trick in Number Theory: by re-arrange ‘1’ as n.[n]-(n² -1), deduce [n], (n² -1) co-prime by the 18CE Bézout Theorem => gcd {[n] , (n² -1) } = 1
Remark: Strange, Bézout is not taught in A- level nor common in UK /Singapore universities.
He used Complex Analysis, what about by conventional method still gets the same answer ?
“Tensor” (张量) was invented for the General Relativity, since the then-existing math had no such algebraic structure.
Math Online Tom Circle 