Math Foundations from Scratch

• Foundations:
  • Some basic foundations
  • Terence Tao's Analysis I book to learn mathematical logic and the construction of the real numbers
  • A geometric course in Linear Algebra to visualize for yourself what is going on
  • Some workshops from Poh-Shen Lo on matrix algebra modeling and understanding the determinant
• Logic:
  • Constructive logic and a seminar on proof theory which is a way to model a proof so you can then check your own work or enter it into Agda
• Practice:
  • CMU Putnam seminar and book Putnam and Beyond to learn strategies and practice them
  • 15-751 CS Theory Toolkit to learn grad research tips, and to better understand complexity

At first, your proofs will just be complete sentences that are a convincing argument to yourself. You don't need to have them verified or a solutions guide or anything. The act of re-reading definitions, experimenting around with the examples, and trying to deduce the logic to write your own proof is how you learn math. Remember when you learned to read and write, at first you just naively scribbled unaware your composition skills were terrible until you took more advanced grammar classes in school, and you began to read other people's writing to compare to your own. Then through experience you got better. This is same deal for proofs, we will eventually see and practice enough of them in the Putnam seminar you will get better at this skill.

Presented here is a definition of a mathematical object called a natural number, as a string or sequence of strokes, which are successors of each other. ie: the successor of a number, is another number.

• YouTube - Arithmetic and Geometry Math Foundations 1 *note these videos eventually get better in quality

• YouTube - Arithmetic and Geometry Math Foundations 2

An interesting explanation of multiplication.

• Laws for addition
  • n + m = m + n (Commutative)
    • Changing the order of the elements.
  • (k + n) + m = k + (m + n) (Associative)
    • Changing the grouping of the elements.
  • n + 1 = s(n) (Successor)
• Laws for multiplication
  • nm = mn (Commutative)
  • (kn)m = k(nm) (Associative)
  • n \(\cdot\) 1 = n (Identity)
• Note the different notation for multiplication:
  • nm
  • n \(\cdot\) m
  • n \(\times\) m
• Distributive laws
  • k(n + m) = kn + km
  • (k + n)m = km + nm

Exercise: Prove why the multiplication and distributive laws work. Since we have no idea what a proof is yet, we can instead try going through some of our own examples.

Identity law: n \(\cdot\) 1 = n.

• We are creating n groups of 1. According to our definition of multiplication |||| \(\cdot\) | would be 4 groups of | or ||||. This is still 4.

Commutative law: nm = mn.

• ||| \(\cdot\) || = || || || or 6. Rearranging this, || \(\cdot\) ||| is ||| ||| which is again 6.

Associative law: (kn)m = k(nm).

• If k = |, and n = ||, and m = ||| and parens expressions are evaluated first:
  • Left side: (kn)m. We first evaluate (kn) as || (2 groups of 1, the identity law), and 2m is ||| ||| (2 groups of 3) for a total of 6.
  • Right side: k(nm): In the parens nm is 2 groups of 3, so ||| |||. k(6) is the identity law, one group of 6. Thus (kn)m = k(nm).

Distributive law: (k + n)m = km + nm.

• If k = ||||, and n = ||, and m = |:
  • Left side: (|||| + ||) is 6, and (6) x 1 is six groups of 1 or 6 according to the identity law.
  • Right side: km + nm: 4 groups of one is 4, plus two groups of 1 is 2 (identity law). Answer is again 6.

• YouTube - Arithmetic and Geometry Math Foundations 3

Wildberger shows another interpretation of multiplication, and walks through the proofs of the laws by rearranging according to the commutative law.

• YouTube - Arithmetic and Geometry Math Foundations 4

Subtraction is the inverse of addition

• If k + m = n then k = n - m
• Note: n - m is only defined if n > m as we are still in the realm of the Naturals and negative numbers/integers aren't yet defined.

+, -, are binary operations and can only do 2 operations at a time

• Laws of Subtraction:
  • n - (m + k) = (n - m) - k
  • n + (m - k) = (n + m) - k
  • n - (m - k) = (n - m) + k
• Distributive laws of subtraction:
  • n(m - k) = nm - nk
  • (n - m)k = nk - mk

Note to remove parens: https://www.themathpage.com/alg/parentheses.htm we can drop the grouping enforcement when using addition (associative law) but must distribute the negative sign, as subtraction is not associative.

• n - (m + k) = (n - m) - k
  • removing grouping: n - m - k = n - m - k

• YouTube - Arithmetic and Geometry Math Foundations 6

A simplified Roman numeral system is introduced to abstract our previous notation of one's so we can manipulate bigger numbers.

• Uses powers of 10
  • 327 = 3(100) + 2(10) + 7(1)

• YouTube - Arithmetic and Geometry Math Foundations 7

Multiplication with the Hindu-Arabic notation:

• Same laws as before:
  • (ab)c = a(bc)
  • a(b + c) = ab + ac
  • (a + b)(c + d) = ac + ad + bc + bd

A different way of hand multiplication is shown, taking advantage of the fact the notation is using the distributive law.

• YouTube - Arithmetic and Geometry Math Foundations 8

Division is repeated subtraction. Laws of Division:

• 1. \(\frac{k}{m}\) + \(\frac{n}{m}\) = \(\frac{k + n}{m}\)

• 2. \(\frac{k}{m}\) \(\cdot\) \(\frac{n}{l}\) = \(\frac{kn}{ml}\)

• 3. \(\frac{k/m}{n}\) = \(\frac{k}{mn}\)

• 4. \(\frac{k}{m/n}\) = \(\frac{kn}{m}\)

• YouTube - Arithmetic and Geometry Math Foundations 9

We're still using natural numbers.

• Def: A fraction, is an ordered pair (m,n) of natural numbers also written m/n.
  • Fractions m/n and k/l are equal precisely when ml=nk.

• YouTube - Arithmetic and Geometry Math Foundations 10

We're still in the natural numbers, which he defined as beginning with 1 so we aren't working with zeros yet (no division by zero corner cases). He defines the operations of addition and multiplication with fractions using the laws of division we already learned.

Note how he sets up the proof. The 'prime' notation (the tick, so c', a', b') represents another variable. You could rewrite this 'Suppose a/b = y/z so that a/z = b/y, and c/d = something/another, so that c/another = something/d does it then follow …'.

What we are trying to prove is ab' = ba' and cd' = dc'. He introduces some extra notation, multiplying both sides by (dd') to help do this.

• YouTube - Arithmetic and Geometry Math Foundations 11

Wildberger proves the distributive and commutative laws using fractions while still in the natural numbers.

• The 3rd edition of Analysis I from WorldCat or LibGen (or buy it, it's an inexpensive book if you get the international version from abebooks or Amazon India)
• The book How to Think Like a Mathematician by Kevin Houston. Most of this won't make sense until we do a few chapter's out of Tao's book, we will read it at the same time.
• The optional book An Infinite Descent into Pure Mathematics by Clive Newstead which provides another view of the same material if you're confused by something, look it up here (or search YouTube for a tutorial)

We begin like we did with Wildberger, basic counting but rigorously defined.

• The five Axioms
  • Axiom 2.1: 0 is a natural number
  • Axiom 2.2: If n is a natural number, then n++ is also a natural number
  • Axiom 2.3: 0 is not the successor of any natural number
  • Axiom 2.4: Different natural numbers must have different successors
  • Axiom 2.5: Let P(n) be any property pertaining to a natural number n. Suppose that P(0) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.
    • We make an assumption about a property P(n) being true meaning you do not prove P(n), you prove P(0) and P(n + 1) are true, and both these form a logical implication meaning that if they are both true, then your initial assumption of P(n) being true holds.

Remark 2.1.10 and the template in Proposition 2.1.11 is probably the clearest definition of proof by induction I've seen. Note the example where no natural number can be 'a half number', because of the increment step n + 1. The base case of 0 (as per Axiom 2.5, and Axiom 2.1) fails if we are using 'half numbers' instead of natural numbers because if zero is a natural number, and any n++ is another natural number, the only next successive number is 1 it can't be 0.5 or -3 or 3/4. The way you should view these axioms about the natural numbers is:

• 0 is an element of N
• if x is an element of N, then so is x + 1.
• nothing else is an element of N

Let's understand Axiom 2.5 by watching a lecture from MIT on proofs though you will end up writing so many induction style proofs on the natural numbers in Tao's book this will become second nature.

• 1. Induction lecture
  • From MIT's 6.042j Mathematics for Computer Scientists
• Read chapter 24 Techniques of proof IV: Induction in Kevin Houston's book

Proposition 2.1.16 (Recursive definitions) in the book Analysis I. Tao uses a function, it's definition can be read in chapter 3.3. Here \(a_0\) is function notation so \(a_0\) is \(f_0(0)\), \(a_1\) is \(f_1(1)\). This proposition is using a new function (a rule that maps from one input to one output) every time, I'm guessing to avoid sequences where they could repeat ie 0, 1, 1, 2, 3..

Let's do an example.

• \(a_0\) becomes 0 (for c = 0)
• \(a_{0++}\) becomes \(f_1(a_1)\) or 1
• \(a_{1++}\) becomes \(f_2(a_2)\) or 2

The proof notes Axiom 2.3 protects the value \(a_0\) from being redefined by one of the functions. His base case for this proof is \({a_0}\) (Axiom 2.3) and he is assuming P(n) which is that the procedure will always give a single value to \({a_n}\). Axiom 2.4 is the inductive step, that this procedure will also give a single value to \(a_{m++}\). All of the Axioms are used to prove this. If this didn't make sense it will when we come back here in chapter 3.5 for an exercise.

Let's understand his example for addition. 3 + 5 should be the same as (((5++)++)++) and adding 0 to 5 should result in 5.

Definition 2.2.1 notice he has a base case (0 + m is m), the hypotenuse "Suppose we inductively defined how to add n to m" which we saw at the beginning of 2.2 where 3 + 5 is defined as incrementing five three times, finally the inductive step, we can add n++ to m by defining (n++) + m to be (n + m)++.

Try writing your own examples as advised in the book How to Think Like a Mathematician

• 0 + m = m
• 1 + m = ((0++) + m) or m++
• 2 + m = ((1++) + m) or ((m++)++)
• 3 + m = ((2++) + m) or (((m++)++)++)
• 4 + m = ((3++) + m) or ((((m++)++)++)++)
• 2 + 3 = ((3++)++) or (4++) or 5.
• 3 + 2 = (((2++)++)++) or ((3++)++) or (4++) or 5.

Lemma 2.2.2 notice first how he is using the exact same induction template from 2.1.11, you will eventually get very familiar with this template. He's using prior established definitions to set up the proofs, notably definition 2.2.1 where 0 + m = m is defined. He assumed that n + 0 = n. Next the inductive step, where (n++) + 0 = n++. No arithmetic was done here, he pointed to the definition of addition from 2.2.1 that (n++) + 0 is (n + 0)++ or n++, so we have n++ = n++ and the proof is done. He used a different variable to not confuse it with the previously established definition of 0 + m = m, since we are now going the other way with n + 0 = n on our way to prove a + b = b + a.

Lemma 2.2.3 Prove: n + (m++) = (n + m)++. We are inducting on n, and keeping m fixed which means we apply the inductive step to n++ instead of m++. Replace all instances of n with 0 for the base case and show the inductive hypothesis holds that 0 plus the increment of some other number, equals that number. Try picking apart this proof yourself to follow it:

• base case:
  • 0 + (m++) = (0 + m)++
  • left side is m++ because of definition 2.2.1
  • right side is m++ because of definition 2.2.1
• inductive hypotenuse: 'Suppose n + m++ = (n + m)++'
• inductive step:
  • n++ + (m++) = (n++ + m)++
  • left side is ((n + m)++)++ because of 2.2.1
  • right side is ((n + m)++)++ because of 2.2.1

Corollary: Tao asks why n++ = n + 1 is a consequence from the two proved lemmas. Compare lemma 2.2.3 For any natural numbers n and m, n + (m++) = (n + m)++ to definition 2.2.1 where we were able to show m++ = m + 1. Compare lemma 2.2.2 to the base case in definition 2.2.1

Proposition 2.2.4 (Addition is commutative) the base case is straight forward. We use the definition of addition, and lemma 2.2.2 to rewrite the successor of n to be (n + m)++ = (m + n)++ and now it is in the form of the inductive hypothesis, so we can use our hypothesis of n + m = m + n to show that (m + n)++ is the same as (n + m)++. Proposition 2.2.6 is similar.

Proposition 2.2.6 (Cancellation Law) uses Axiom 2.4 which states 'Different natural numbers must have different successors; i.e., if n, m are natural numbers and n = m, then n++ = m++. Equivalently, if n++ = m++, then we must have n = m'. So (a + b)++ = (a + c)++ is = a + b = a + c by the line if n++ = m++, then n = m in axiom 2.4. We dropped down from the successor by using axiom 2.4 and can now use our inductive hypothesis of the cancellation law for a to show b = c and close the induction. .

Lemma 2.2.10 has an exercise where induction is suggested but this is actually much easier to prove using contradiction (See Kevin Houston book). Always read the comments on Tao's blog if you're stuck: "You can modify the statement of the exercise to an equivalent statement which makes sense for a=0. (The statement will probably look something like “For any natural number a, either a=0, or else there exists exactly one…”.)" If you see this stackexchange answer they use the peano axioms of 2.2 existence and 2.4 uniqueness. We can also try the chapter on contradiction proofs in Kevin Houston's book or the appendix of Analysis: Suppose for the sake of contradiction that that there exists two different natural numbers with the same successor. Then a = m++ and b = m++. But then by Axiom 2.4 we have a = b, a contradiction.

Definition 2.2.11 n is greater or equal to m if n = m + some other natural number. If that number is 0, then the equality holds, if that number is 0++ then the inequality holds.

Proposition 2.3.4 follow along with definition 2.3.1 of multiplication where (n++) x m = (n x m) + m also recall that n++ is n + 1: (n + 1)m so nm + m

• left side:
• a(b + c)++)) is a(b + c) + a
• because (m(n++)) = m(n) + m
• right side:
• ab + a(c++) = ab + ac + a
• ab + ac + a is in the form of the inductive hypothesis so a(b + c) + a

Remark 2.3.8 the contradiction is that ac = bc by corollary 2.3.7 when fed into the trichotomy of order can only result in a = b. This is a scheme to produce a temporary notion of division/cancellation.

• Pythagoras' Theorem 1 and Pythaoras' Theorem 2
  • A geometric proof of original Pythagoras theorem
  • A proof by contradiction why the square root of 2 is irrational
  • Part B shows classical cos/sin/tan trigonometry using chords

• Greek Geometry 1 and Greek Geometry 2
  • Euclid's book the Elements
  • Part B covers different planes, ellipse/conic geometry

This is the geometry we will be doing shortly that allows for computational Euclidean geometry. We don't have a definition of polynomials yet but you can probably understand most of this lecture anyway.

• Analytic Geometry
  • Cartesian coordinate plane explained
  • Part B covers periodic continued fractions method to estimate irrational numbers

Abstract geometry, project from one plane to another like 2D to 3D or into another field to reduce complexity.

• 3D Projective Plane
  • Shows you different perspectives when looking at objects
  • Lay the x,y cartesian plane flat, add a z dimension like a flag pole, add another flat cartesian plane in z, project lines from x,y into z

Even more abstract geometry, construct a new plane by slicing a sphere and studying a hyperbola. Hyperbolic geometry seems better suited for deep learning as it has 'more capacity'. Maybe we will end up back here projecting our data into a hyperbolic plane.

• Non-Euclidean/Hyperbolic Geometry
  • Multiple slicing planes/perspectives

Geometry of space curves, osculating plane

• Differential Geometry
  • Abstractions on top of abstractions
  • There's even more abstract geometry

Flip through sections 1.3.1, 1.3.2 and 1.3.3 Review, Redo, Renew from this draft book used in CMU's 15-151/21-28 course. The author has a list of errata on his page. You will get a crash course in some quick arithmetic, solving linear equations and polynomials. There's a few interesting methods here you may not know about like how to variables from equations. Wildberger covers much of this material too for example Problem 1.3.8, an infinite nest is also in a continuous fractions lecture from the History of Mathematics playlist: \(x = \sqrt{2 + x}\)

If you're interested there is a summary of Algebra here that was originally made for non-english speaking university students. Tao will also define everything here as we progress through his analysis texts.

Some optional mostly 10 minute lectures on algebra basics, such as how Pascal's array/Khayyam triangle/Yang Hui's triangle encodes the binomial theorem, this series sets up understanding algebra and later calculus and combinatorics.

• Introduction to Algebra
  • Wildberger notes there is a subtle change in variables as you get more sophisticated with algebra
• Baby Algebra
  • The basics of solving equations
• Quadratic equations I and part II
  • Completing the square
  • The quadratic formula
• How to find a square root
  • Manual square root algorithms
• Number patterns I
  • Table of differences, summation notation introduced
• Number patterns II
  • Table of differences geometric interpretation
• Euler and Pentagonal numbers
  • Recursive partition function
• Algebraic identities
  • Common identities
• The Binomial theorem
  • Coefficients are related to paths on a two dimensional array
• Binomial coefficients
  • N choose K intro to combinatorics
• Trinomial theorem
  • 3D Pascal's tetrah

Before we do trig, let's learn what \(\pi\) is. Wildberger talks about the history of pi such as Archimedes interval for pi, Newton's formula, a hexadecimal version of pi, his own formula. The end of this lecture he claims pi is some kind of complex object we still don't understand. At this stage we only need to know it's a constant, or the applied approximation of 22/7 or 355/113.

Now that we know trig functions, watch this basics of calculus crash course which was designed for non-english speaking students to learn math notation (with Chinese subtitles). Note in the last few examples how the derivative works d = \(nx^{n-1}\) so in the examples the derivative of \(y = \frac{-g}{2}t^2+v_0t+y_0\) is (\(2\cdot\frac{-g}{2}t^{2-1}+ 1\cdot v_0t^{1-1}\)) or \(-gt + v_0\) you take the exponent power, multiply the coefficient of the base, and subtract the power, so the derivative of \(-gt^1 + v_0\) is (\(1\cdot-gt^0 + v_0\)) which is -g, as we'll see later any terms without a variable (in this example t) cancels out or as Wildberger mentioned are so small so they are discarded.

Summary: if you know the acceleration, then you integrate to find the velocity, then integrate again to find the position. If you know the position, you take a derivative to find it's velocity, and take another derivative to find the acceleration. One function tells you how another function is changing. These are also higher order functions we learned in CS019, where a function produces another function.

Try reading the first chapter from Gilbert Strang's calculus and see if you understand it. His book was designed for self-learning, so that first chapter assumes little background and builds up this intuition of one function that tells you about another. There is also a lecture for this chapter and a self-study guide that acts as a recitation.

There is a lecture on the history of sets/functions, explaining ordinals, cardinals, various schools of logic, and a second lecture about the history of set theory with Godel and Turing.

This chapter on set theory in Analysis I relies on the fact that you've already read the appendix on logic and have an idea what implies means or disjunction. Now that we have done some proofs ourselves, and the basics of trigonometry we can finally understand some of the examples he uses in this appendix. Tao makes a point that this chapter is meant to be read and understood not studied and memorized, and thankfully he is not bombarding us with piles of confusing logic notation that you normally see in other 'how to do proofs' books. At the same time you read this appendix, read How to Think Like a Mathematician by K. Houston it will clear up any cryptic definitions like how implication works.

A.2 Implication seems confusing on first read, it's modern language use relating to casuality is different from the logician's use which is explained here. The only permutation where implication is false is if p implies q and p is true, but q is false. See the Kevin Houston book for more details.

An optional short series on the history of various schools of logic that led to our present state.

• Aristotle and deduction - Brief history of logic MF 251
  • Origins of modern logic
• Stoics and other thinkers - Brief history of logic MF 252
  • Implication discussed
• Islamic/Medieval logic - Brief history of logic MF 253
  • Modal logic origins
• Leibniz to Boole - Brief history of logic MF 254
  • From deduction to induction
  • Notation for relations
  • Boole's book Analysis of Logic
  • De Morgan's laws

Proposition A.2.3, confusing the hypotenuse "Suppose that…" by assuming the conclusion. We always assume the hypotenuse, and prove the conclusion. Proposition A.2.6, proof by contradiction, we know that when pi/2 (90deg) the sine function is 1, because remember it is the y axis of the unit circle. Tao also makes a note here to not use confusing logical notation in proofs.

Remark A.5.1 is in the above history of logic lectures about Aristotle logic.

Exercise A.5.1

• a) For every x++, and every y++, y*y = x or x is the square root of y. Is this true? Square root of two is the obvious candidate for claiming this is a false statement since what y * y = sqrt(2).
• Think about the rest, using the K. Houston book as you go to find a contrapositive

A.6 the point of the examples is to make a point about arguments that use outer variables depending on inner variables being false. We can come back later and reread this section when we learn more concepts like limits, indeed we will likely have to reread this entire appendix many times over while writing future proofs.

We are back to reading Tao's Analysis I and still in the realm of the natural numbers. Example 3.1.2 is similar to a nested list such as [list: 1, [list: 1, 2], 100] is length 3, even though there is a nested list overall it still only contains 3 elements. Remark 3.1.3 is in the above historical lectures on set theory. We are referred to Appendix A.7 Equality to read about the substitution axiom which you should have already read.

Lemma 3.1.6 if set A is equal to set (empty), then if x is in A it must also be in set empty, a contradiction, so A must be a non-empty set is how I interpret the lemma, which is a way to say that if an empty set exists, there must also be a possibility of a non-empty set. Remark 3.1.8 Cardinality if you watched the Wildberger History of Mathematics lecture linked above assigns a value.

Note the errata for Definition 3.1.4 from Tao's site: "Definition 3.1.4 has to be given the status of an axiom (the axiom of extensionality) rather than a definition, changing all references to this definition accordingly. This requires some changes to the text discussing this definition. Firstly, in the preceding paragraph, “define the notion of equality” will now be “seek to capture the notion of equality”, and “formalize this as a definition” should be “formalize this as an axiom”. For the paragraph after Example 3.1.5, delete the first two sentences, and remove the word “Thus” from the third sentence. Exercise 3.1.1 is now trivial and can be deleted." So this is now Axiom 3.1.4, but the definition remains the same.

Exercise 3.1.2 prove emptyset, {emptyset}, {{emptyset}}, and {emptyset{emptyset}} are all not equal.

• If \(\emptyset\) = {\(\emptyset\)} then by Def 3.1.4 \(\emptyset\) \(\in \emptyset\) a contradiction to Axiom 3.2
• Same reasoning as above for \(\emptyset\) is not equal to {{\(\emptyset\)}} or {\(\emptyset\){\(\emptyset\)}}
• {\(\emptyset\)} is not equal to {{\(\emptyset\)}} because [list: 1] is not equal to [list: [list: 1]]. The first has element 1 in it, the second has element [list: 1]. More precisely if {\(\emptyset\)} = {{\(\emptyset\)}} then by Def 3.1.4 every element of {\(\emptyset\)} (it's only element is \(\emptyset\)) must be in {{\(\emptyset\)}} (it's only element is {\(\emptyset\)}) and vice-versa, so claiming {\(\emptyset\)} \(\in\) {\(\emptyset\)} which contradicts Axiom 3.3, where y \(\in\) {a} iff y = a. In this case y must = \(\emptyset\) to be in {\(\emptyset\)}.

Remark 3.1.21 clarifies the above exercise how {2} is not in {1,2,3} but 2 is in {1,2,3}. The rest of this chapter is straight forward definitions of set operations, better than the usual treatment of venn diagrams. Remark 3.1.26 distinct is defined as there are elements of one set which are not elements of the other so \(\emptyset\) and \(\emptyset\) are not distinct because neither can have elements. In Proposition 3.1.28 notice how any operation on two sets results in a new set, like a higher order filter function over a list results in a new list.

Example 3.1.32 looks a lot like programming syntax. There seems to be an order of evaluation, applying the axiom of specification first to create a new set, then applying the axiom of replacement so working right to left in {n++ : n \(\in\) {3,5,9); n<6}

Example 3.3.3 if you forgot range notation. Remark 3.3.5 clears up some notation for functions notably a sequence subscript also denotes a function. Example 3.3.9 asks why 3.3.7 asserts that all functions from empty-set to X are equal, because they have the same domain (nothing) and range.

Our first proof in this chapter is in Lemma 3.3.12, and if we can go by all the previous chapters, this will be our template for the exercises. It also doesn't make sense, so I checked the errata page and sure enough there is errata for this lemma depending what version of the book you're using.

3.3.14 See this tutorial. Write your own examples for this definition. If two distinct elements of A map to the same element of B, this is not injective. However if those two A elements are equal to each other then it is injective:

• A = {a, b, a} (function input)
• B = {1, 2} (output)
  • f(a) = 1
  • f(b) = 2
  • f(a) = 1
    • but {a} = {a}, so a = a and this function is injective.
• A = {b, c}
• B = {2}
  • f(c) = 2
  • f(b) = 2
    • but {c} doesn't equal {b} so this not injective

3.3.17 Onto functions, another tutorial and for Remark 3.3.24 here is a tutorial for bijective inverse functions

Watch this starting @ 36:00, from 41:00+ he shows how he solves problems step by step, trying small cases to understand the problem better and poking at the problem enough to finally figure out a proof. You can check your own proofs too, going through Tao's mathematical logic appendix and seeing if the opposite of your argument holds, and double checking all your assumptions 'is there a definition of X or did I just assume it existed'.

So tl;dr relax and just keep writing because the practice of flipping back through the text and rereading definitions and theorems, tracing the logic and seeing how one definition helps define another, and putting this in the form of notes (a proof) is how you learn. This is why none of these books contain solutions because you're supposed to figure it out yourself.

Chapter 3.4 of Tao's Analysis I. Example 3.4.5 makes more sense if you read Example 3.4.6 too, f^-1({0,1,4}) is the set of everything that could result in {0,1,4} if f(x) = x², so {-2,-1,0,1,2} and in Example 3.4.5, the set of everything in the natural numbers that can result in {1,2,3} if f(x) = 2x, is only f({1}). Axiom 3.11 (Union) if A = {{2,3}, {3,4}, {4,5}} and Ua = {2,3,4,5} why? Because it's {2,3} U {3,4} U {4, 5} you are performing union on every set, this is notation to avoid having to write out an enormous page of union operations. Search youtube for indexed sets. The power set you can choose which subsets you want to add, again search for this on youtube.

The exercises, we've already done similar exercises in set theory there's nothing new here. Reminder that millions of discrete and applied math students have already done similar exercises so you can look up proofs of function inverse images anywhere. I did enough of the exercises so I was sure I understood these set and function operations otherwise move on.

Another topic with plenty of youtube tutorials. Definition 3.5.7 notation for product is introduced, it looks complicated but all it's doing is specifying an index (i is greater or equal to 1, and less than or equal to n) and repeated cross product. The exercises some of the solutions are walked through on youtube.

The cartesian plane itself is a cartesian product of R x R, you have tuples (x, y) or (x, y, z) so R x R x R. In programming the type int * int is the set Integers x Integers, anything that isn't a cross product of the integers with itself will not type check.

A good tutorial. Tao is using #(A) notation which means finite cardinality, while |A| cardinality notation means infinite. There exists errata for the exercises specifically 3.6.8. The last exercise the pigeonhole principle if set A has greater cardinality than B, then f(x) A -> B can't be injective, since there would be two elements in A that point to a single element in B.

If you want you can re-do what we've done so far to get better at this material by programming sets using the book Discrete Math and Functional Programming. Everything we covered such as peano axioms, inductive proofs, cardinality, function images and even converting math notation into programs is covered and highly recommended.

This is unique in that each linear algebra topic is setup with the problem it's trying to solve, how the special case of that problem can be solved, then how you would go about creating and proving a general case. The determinant is built up from scratch. The exercises are largely proofs of the lecture content presented at the end of each lecture. It frequently goes into much more advanced material you won't find in any other linear algebra course which is possible since we're just using a simple affine plane, and working with vectors. The idea is full understanding of the subject using visual geometry so later when we read piles of sums and notation for matrix manipulation we can understand what they are doing. This course teaches you to become 3blue1brown where you can make your own visualizations.

• Introduction - Wild Linear Algebra A1

He constructs the affine plane from scratch. The vector (8/3, 10,3) you need to full screen to better see exactly where they are. From B, you move right 2 lines, and 2/3 so \(2 + \frac{2}{3}\) which is 8/3. From there you move up 1/3, and 3 so \(3 + \frac{1}{3}\) which is 10/3. Reasons for using the affine plane are you do not impose anything on it artificially like the origin (0,0) in the Cartesian plane, you can do projections easily, and it's construction is simple, since only parallel lines.

The biggest a takeaway here is what vector direction is, what a basis vector is and how you can combine them to form larger vectors, and how to translate basis vectors to suit different perspectives. The exercises at the end the delta symbol will return in the lecture on area and the determinant.

In this optional related lecture he shows how ancient people made a ruler using just parallel line geometry, which leads to modelling the rational continuum. Some more optional (short) videos is this from 3Blue1Brown on what a vector is, and this presentation of basis vectors and what a span is. Change of basis is video 13 in the 3BlueBrown playlist but the explanation here by Wildberger in the first lecture I find is better.

• Geometry with Vectors - Wild Linear Algebra A2

All this vector arithmetic will come up later in other courses, at least now you will be somewhat familiar with it. Subtraction means going in the opposite direction, so if you're at the head of one vector and going to another vector's head, you are subtracting. The generalized affine combination where coefficients add up to 1 will come up again.

The proof for diagonals of a parallelogram bisect each other, note how quickly this can become a pile of symbols.

• let lambda = mu in: lambda = 1 - mu
  • if lambda = mu, then moving mu over to the other side means 2lambda
  • 2lambda = 1
  • Divide each side by 2 to isolate lambda
  • lambda = mu = 1/2

@38:00 or so solving linear eq with 2 unknown variables we already covered in the first lecture or review Crash course in solving equations & polynomials.

Since we just did Physics in the software workshop (CS019 Differential systems) let's learn about Archimedes principle of the lever, accelerations in the grid plane and Newton's laws. This will come up again, note the labs here for a course in applied math specifically Lab Manual vol 2, writing python fuctions that perform Barycentric Lagrange interpolation. This is used to study air pollution by approximating graphs.

The fourth lecture in the series to lead up to determinants. You probably recognize ad-bc from the first lecture, which was one of the exercises. He uses different notation for bi-vectors since it represents the union of vector a and b. The torque example you have probably seen somebody using a long pipe over a wrench in order to increase the torque without needing increased force, now you know why increasing the area of the wrench increases the torque. All 3 physics examples can be modelled with a parallegram/bi-vector. The bi-vector rules (ie: adding a scalar to one side of a bi-vector doesn't change the bi-vector) will make sense around 35:50 when he produces a picture proof showing how moving things doesn't change the bi-vector. At around 38:00 we once again get the formula ad-bc, the signed area is a scalar of two basis vectors. 45:50 introduces three-dimensional affine space.

Now that you've seen this model of area using bi-vectors, see this 10 minute presentation by 3Blue1Brown which will fill in some more details, like why the signed area can be negative if you didn't already figure it out. The exercises are conceptually easy, you don't have to do all of them but the whole point of linear algebra is to take these geometric concepts and do them in algebra.

Lecture 5. Now we get a full intro to matrix and vector notation, and matrix operations, this is normally where linear algebra courses start. I like the fact that he built up the change of coordinates first so we aren't just presented with a meaningless array of numbers.

Exer 5.1 if you make a B matrix with e, f, g, h and then multiply AB, take the resulting matrices determinant. The piles of letters should equal det(A) x det(B) after you do some commutative rearranging, this is a 10 minute exercise.

Lecture 6. I appreciate he starts all these lectures with a review, that makes them mostly self contained. This is something I haven't seen before, explaining all the arithmetic of 2x2 and higher matrices using just a 1x1 matrix. The inverse of a matrix is now simple to understand. He walks through some proofs of matrix laws like the associative law (AB)C = A(BC). The standard cartesian coord system is used here for examples of transformations like dilations and shear. A dilation is growing or shrinking, like your eye being dilated.

The end of this lecture: how do you visualize an arbitrary transformation? The next lecture expands on this intuition.

Lecture 7. This tells you how to visualize transformations T(e₁), T(e₂) are the columns of a matrix, so you can look at it and see how it is being transformed. He briefly introduces an alternate framework of rational trig so we don't have to use those sin/cos functions to figure out a rotation. There is a book and entire playlist for rational trig, if you want to learn it write your own software library for rational trig as you go.

End of the lecture introduces non-linear approximations, using differential calculus, to set up doing calculus in higher dimensions. The exercises you become 3Blue1Brown, able to visualize transformations yourself.

Lecture 8. 3 dimensional linear algebra. Pretty good setup explaining the inverse of a 3x3 matrix, most books just give you a formula. We find out why delta is 1/delta. Transpose is also explained better than I've seen before. Reminder that these really complicated introductions all have an off the shelf formula, but he's showing us how they are derived fully so you understand the formula. 3x3 matrix multiplication is explained and pretty simple, explains why B(A(vec)) is the same as (BA)vec. If you're also doing the AI workshop, in 15-388 it's claimed B(A(vec)) is preferable in terms of complexity of running time to (BA)vec.

• YouTube - Arithmetic and Geometry Math Foundations 12

Wildberger's definition is similar to Tao's, he walks through a proof of a-(b-c)=(a-b)+c

Chapter 4 of Analysis I by Tao. Definition 4.1.1 is exactly the definition from the above Wildberger foundations video. Lemma 4.1.3 proof you can finish most of it yourself without reading the whole proof. Lemma 4.1.5 Tao notes the same concerns Wildberger expressed why they both used the 'a less b' style of definition instead of just using Lemma 4.1.5 as the definition of integers, because of the many different cases that would be needed. Tao mentions this definition of the integers forms a commutative ring because xy = yx.

• YouTube - Arithmetic and Geometry Math Foundations 13

This definition is similar to 4.2 in Analysis I. The inverse law, write your own example, the rational number 4/1 inverse is 1/4 and 4/1 x 1/4 is 4/4 or 1. A rational number is (a-b)/(c-d) according to it's definition but we write them as p/q and in some cases just p if q = 1.

• YouTube - Arithmetic and Geometry Math Foundations 14

The rational number line sits on [0, 1], so you don't end up with a zero denominator which isn't defined. A rational number is a line through the origin [0,0], and an integer lying on point [m,n]. Ford circles are an example of the massive complexity of the rational numbers, which are expanded upon in the next lecture.

• YouTube - Real numbers and limits Math Foundations 95

This is about 35m long but worth the watch if you want to see what a convex combination is, learn about rational intervals, and a more complete description of Ford Circles set up with Farey sequences. This will help us understand what a sequence is later in Tao's book. The overall takeaway of this lecture is that the smaller the interval you take on the rational line, the more complex the numbers become with enormous denominators meaning it has a very deep structure.

The next videos, #96 and #97 about Stern-Brocot trees and wedges are highly recommended and can also be found in the book Concrete Mathematics. In Lecture #97, which we can now understand that we know basic linear algebra, he takes a visualization on the plane of the Stern-Brocot tree and shows it also can be used as a geometric interpretation for the rational numbers, it's kind of crazy when you see it. The 'fraction' 1/0 is also explained.

Tao's Analysis I chapter 4.2, definition 4.2.1: 3//4 = 6//8 = -3//-4 we know from Wildberger's visualization of the rational numbers, these numbers all lay on a line. Lemma 4.2.3, in the proof of addition getting rid of +bb'cd is a simple case of subtracting it from both sides, cancelling it out leaving ab'd² = a'bd² and he can divide both sides by d because d is known to be non-zero, since it's in the denominator of a/b + c/d.

To recap we defined the natural numbers, embedded them in the integers (a - b) to define integers, then embedded integers inside the rational numbers where the arithmetic of the integers is consistent with the arithmetic of the rationals. The rationals are a field because of the last identity in 4.2.4 the inverse algebra law. Tao's example proof is similar to the proofs we've seen Wildberger do, convert the letters into integer // integer form, apply law, commute the results together so that they are the same.

The next chapter in Tao's Analysis I uses decimals to represent tiny positive quantities (epsilon) so we should probably know what decimals are.

Wildberger has a basic introduction lecture using natural numbers that assumes you know nothing about exponents and reviews the hindu-arabic notation, shows how to convert a decimal to a fraction. The second lecture introduces rational number decimals and their visualization.

His definition is a decimal number is a rational number a/b, where b is a power of 10 (hence deca/decimal), then shows various examples how you can manipulate rational numbers like 1/2 to be 5/10, or 0.50, we also find out why 1/3 is not a decimal number. In binary, it's a base-2 system, so the only prime factor is 2. If you load up any language interpreter and try and add 0.1 + 0.2 you will get surprising results, typically this number: https://0.30000000000000004.com/

The examples are pretty good, showing how the hindu-arabic encoded powers of 10 drop down by the divisor and result in a decimal. The visualizations are for later lectures showcasing the complexity of bigger and bigger decimals. At 36:00 arithmetic w/decimals is covered, showing how the rational number and decimal number arithmetic methods are the same so you can just stay w/rational numbers if you want converting back to decimals at anytime.

Def 4.3.1 you apply that def to the rational number, so if x = -3, the absolute value is -x so -(-3) which of course makes it a positive rational number, as seen in the example of 4.3.2 where the absolute value of 3 - 5 = 2.

Def 4.3.4 that symbol is epsilon, meant to be a tiny quantity he defines as greater than 0. The example 4.3.6 if two numbers are equal, and their difference is zero, then they are 'epsilon close' for any positive epsilon, since their difference is not bigger than any positive epsilon.

Proposition 4.3.7 recall that all these definitions are for d(x, y) or d(z, w) so x - y or z - w. His proof he moves xz to one side, so it becomes negated (subtract from both sides). He then manipulates |az + bx + ab| according to previous definitions to get |a||z| etc., and then uses the various propositions in 4.3.7 to rewrite them as eps|z| + delta|x| + eps(delta). This will make sense after we attempt to prove the rest of the propositions.

Rest of this chapter covers exponents and their properties.

Proposition 4.4.1 there is an integer n, and a rational number x, such that n <= x < n + 1 so sandwiched between integers 2 and 3 are rational numbers such as 25/10 or 100/40 etc.

Proposition 4.4.3 there is always some rational you can squeeze in between two other rationals as we've already seen in Wildberger lectures. 1/800 < 1/500 < 1/100 etc. If you want put in examples for his generalized proof to better understand it like x = 2, y = 3 and z = (2 + 3)/2.

Proposition 4.4.4 famous proof by contradiction there does not exist a rational number x for which x² = 2

Proposition 4.4.5 is basically saying for any epsilon bigger than 0, there exists a rational number that when squared it's less than two, and less than x + epsilon. Remember there is an infinite amount of rational numbers between 0 and 2 like trillion/1000 billion = 1. 1² is 1, if epsilon is 1 so that x + epsilon = 2, then (x + eps)² is (1 + 1)(1 + 1) and bigger than 2. He asks why ne (e = epsilon) is non-negative for every natural number n in (ne)² and the parens grouping ne and squaring it obviously make it positive ie (-2)².

Tao mentions Dedekind cuts will be avoided, and the theory of infinite decimals has complexity problems so also will be avoided. Wildberger has numerous lectures on both theories if you're interested the most detailed ones are in his 'Famous Math Problems' playlist. Wildberger makes the point that you will only ever find the cherry picked example of the square root of 2 for Dedekind cuts in any undergrad material because of the mind boggling complexity involved trying to partition a set for pi²/6 and other irrational numbers.

The infinite descent exercise Tao has basically handed the solution to us, if you fix any sequence of natural numbers then by their definition there are finitely many numbers from 0 to your fixed point, so you can't choose infinitely many distinct natural numbers less than your fixed point so infinite descent does not exist for the naturals because you'll run out of them. Integers you can infinitely descend to negative infinity and of course rationals we know you can infinitely descend with bigger denominators until you approach zero or negative infinity.

• YouTube - Real numbers and limits Math Foundations 106

Watch the first 20 minutes of this lecture and you'll see what the notion of an epsilon displacement is. Note that Tao will define all these things for us later, but at least you'll have an idea of what epsilon is. The rest of the lecture is excellent covering a different definition of limits but optional.

• YouTube - Real numbers and limits Math Foundations 94

Watch from 2:20 to 16:00, he shows the 2D geometric interpretation of Cauchy sequences, and how they are interpreted as 1D on the real line, the idea of 'converging sequences'.

• YouTube - Real numbers and limits Math Foundations 111

He explains the triangle inequality around 8mins, we just read about this in chapter 4.3 of Analysis I (and is one of the exercises).

• YouTube - Real numbers and limits Math Foundations 112

An excellent lecture why we even need these complex limit and real definitions in the first place, showing how limits make many subjects in analysis much easier. Archimedes bracketed measurement is covered when he tried to figure out the approximation of pi. We also get an introduction to transcendental functions as a geometric series, something we will see in Analysis II by Tao in detail. The next lecture in the series is optional but Wildberger presents an alternative definition of real numbers, using Archimedean style of nested intervals where equality, algebra laws, order etc are all trivial to prove.

Def 5.1.1 the notation outside the bracket is n = starting index, and the infinity symbol is the length of the index.

Example 5.1.2 reminds us of Def 4.3.4 d(x,y) = |x - y| \(\le \epsilon\) so if x and y are 2, then 2 - 2 = 0 and every single episilon is valid since x - y will always be less than epsilon.

Def 5.1.6 if you watched the Wildberger lectures on What Exactly is a Limit then you can see what Tao is setting up here with epsilon-steadiness, there is some index in the sequence where the entries past that index are epsilon-steady. Example 5.1.7 the sequence 10,0,0,0.. d(10,0) is 10, so this sequence is not eps-steady for any epsilon less than 10, but if we choose N past 10, then all the remaining sequence is 0,0,0.. and d(0,0) is 0 - 0 so as per Def 4.3.4 will always be less than any epsilon so 'eventually eps-steady for every eps > 0'.

Def 5.1.8 if you saw the above Wildberger lectures on Cauchy sequences this is defining a convergence so any sequence that eventually becomes epsilon-steady, is a Cauchy sequence. Example 5.1.10 depending on your edition there's errata, in mine (3rd edition Springer from libgen) the example makes no sense.

Prop 5.1.11 proof he cites the fact that there always exists some rational number you can squeeze in between numbers and chooses 1/N noting that no beginner would come up with this proof and it is merely staging in preparation of learning what a limit is. Later in Example 5.1.13 he notes Prop 4.4.1 can again be used to prove the infinite sequence 1, -2, 3, -4.. has no bound, I would guess because of the 4.4.1 statement 'there exists a natural number N and rational x such that N > x ie: no such thing as a rational that bounds all the natural numbers so an infinitely increasing and decreasing sequence can't be bounded.

Exercise 5.1.1 since we declared it a Cauchy sequence, you can point to the definition of Cauchy sequences being an object that eventually becomes epsilon-steady, so Lemma 5.1.14 proves the finite part of Cauchy sequences up to epsilon-steady are bounded, and obviously epsilon-steady means it's bounded since you can subtract two indices and they will always be less than epsilon if this is a Cauchy sequence and epsilon-steady, or 1-steady, 2-steady, 1billion-steady. So both are bounded are ergo all Cauchy sequences are bounded using his definition of episilon-steadiness.

Prop 5.2.8 is the proof that 0.9999… = 1, and again prop 4.4.1 is brought out to claim we can always choose any rational number we want to make 2/N <= epsilon. If you follow this proof:

• (1 + 10^-n ) - (1 - 10^-n )
• 10^-n + 10^-n = 2 x 10^-n
• 10^-N is \(\le\) 1/N because of the chapter on exponents, if N = 3 then 10^-3 is 1/3x3x3 which is smaller than 1/3
• Now choose an N that satisfies the requirement that 2/N \(\le\) \(\epsilon\) and it will automatically be true that it also satisfies 2 x 10^-N \(\le\) \(\epsilon\)
  • Prop 4.4.1: you can always choose some rational to shove in there that will satsify the req

Only two exercises here, simple point to definition pretend you are a lawyer writing an argument and convince an invisible jury through explanation why it's true an equivalent sequence of rationals is a Cauchy sequence if and only if one of them is a Cauchy sequence. There's no need to pull out any fancy techniques here we will learn all those later.

Tao has one of the few books anywhere that actually defines the real numbers. Just like understanding the constructions of rationals and integers helped us understand their operations better, understanding the construction of the reals should help us better understand crazy abstractions like x^e or irrational square roots.

Def 5.3.1 the formal limit of a Cauchy sequence, meaning the number a sequence is converging towards, is a real number. Every rational number is also a real number, for example if we choose 5, then a Cauchy sequence converging to 5 like 4.99999… is bounded by 5 and is a real number. The notation is: x is a real number if x = the limit of a Cauchy sequence a_n, as the sequence goes from index n to infinity so a₁, a₂, a₃ …

The rest of the chapter covers the same definitions we've already seen except now we are working with a different object, but since it's construction is entirely made up of rational numbers in a sequence almost all the laws are the same. Division by zero not being allowed for real numbers is explained using the sequence 0.1, 0.01, 0.001.. is actually converging towards 0, and it's inverse 1/0.1, 1/0.01, 1/0.001 is 10, 100, 1000… an infinite unbounded sequence therefore not a Cauchy sequence, therefore not a real number, so division by zero is undefined.

I am just casually reading the proofs in this chapter, Tao has simplified it for us with his epsilon-close 4.3.7 propositions and his definition of equivalent Cauchy sequences. This is a much more intuitive and clear You have to unravel all the abstractions like we did before with rationals and integer definitions but they make sense thanks to his prior staging for example the proof of Lemma 5.3.17 is surprisingly easy.

The proof of Lemma 5.3.6 was already explained by Wildberger here if you're wondering what epsilon/2 means.

Looking at the exercises, I'm guessing to prove 5.3.5 we show the reciprocal of 1/n is unbounded so we must be working with a sequence not bounded away from zero. In prop 5.1.11 Tao proved the 1/n sequence is a Cauchy sequence, and since all Cauchy sequences are bounded you can also show it's bound is 0. There is worked out solutions for this chapter here but remember there is no single proof answer, try to prove it a different way.

I thought this chapter would be tedious since we've already done ordering, but it was interesting to go back to prior definitions and see how they work when cast into this Cauchy series abstraction. Prop 5.4.8 if 5 > 4 then 1/5 < 1/4.

Prop 5.4.9 'non-negative reals' means 0 or positive, while positive reals do not include 0. Closed I take to mean a set is closed iff the operation on any two elements of that set produces another element of the same set, in other words adding two non-negative real numbers produces another non-negative real. The explanation of the proof is in the comments here but Tao tells us we'll see this more clearly in future chapters.

Prop 5.4.14 we already know from previous proofs and Wildberger lectures that there is an infinite amount of rational numbers you can squeeze in between two other numbers.

Now we learn analysis, the sup(E) least upper bound property that will come up in every single future chapter.

Example 5.5.7 the empty set doesn't have a least upper bound because of prior definitions where you can always find a rational number (which are also contained in the real numbers) to act as an upper bound to nothing like 1/trillion, 1/septillion, etc.

The proof for Theorem 5.5.9 is huge requiring going back to look at the Archimedean property again and some exercises. Wildberger has broken down this proof somewhat showing the gist of what is going on but the last part of the proof that the upper bound S is less than or equal to every upper bound of E is easy to follow.

Remark 5.5.11 we get the answer to 'can you add negative infinity with positive infinity and get zero' which is no.

The proof of prop 5.5.12 makes sense until you get to that inequality, wondering where he got x² + 5e, which is the Archemedian property. This is basically saying x is the least upper bound, x plus some tiny displacement (epsilon) squared is less than or equal to the Archimedian property of x² + 5(displacement) since if epsilon is really small, squaring it means it will be even smaller. He derives a double contradiction that x² is bigger than 2, and x² is less than 2, so x² must equal 2 and magically we have a number that solves x² = 2 which is not rational.

Exercise 5.5.1 you can type into youtube 'proof of infinum' and get a dozen tutorials or you can go through Prop 5.5.8 and Theorem 5.5.9 to change everything to prove a least lower bound which gives some more experience with the Archmedian property which all these proofs seem to rely on. The rest of the exercises 'draw a picture' think of Wildberger with his within epsilon/2 picture in earlier lectures.

Finally we learn square roots. Depending on your copy of Tao's book there's some minor errata here to look up. Note in the proof of Lemma 5.6.8 he's using substitution, if \(y^a = x^{1/b'}\) then we can substitute them both for any occurences, such as \((y^{a})^{a'}\) = \((x^{1/b'})^{a'}\)

Exercises 5.6.2 and 5.6.3 are simple and worth doing, using substitution to manipulate exponent laws. Exercise 5.6.3 that's the square root of x², so x or using Lemma 5.6.6(g) or Lemma 5.6.9(b): \((x^{2/1})^{1/2}\) is \(x^{2/1 \cdot 1/2}\) which is x¹ or x.

You now know what a real number is, sort of. It's an infinite sequence, with a bound meaning it converges to a limit and that bound is the real number. So for example .9999… just keeps growing but it never gets bigger than 1, so it's limit/bound is 1 and therefore the real number 1. The proofs make sense but the idea of everything being a sequence wasn't really explained except that it is a convenient abstraction for introducing the least upper bound, and therefore a limit. We didn't cover the real decimals appendix either which we'll have to look at after reading chapter 6.

We're watching this lecture: Three dimensional affine geometry | Wild Linear Algebra 9. He actually has a physical model of 3D affine space and shows how it projects to 2D. 3D (x, y, z) coordinates are explained with an interesting question 'how to model space?'. \(A^3\) space and \(V^3\) space are defined, insersecting and parallel planes are illustrated.

As typical with Wildberger lectures this goes far into more advanced material like Mobius bands to model all the lines in the plane, nobody ever showed me this in my linear algebra classes.

Lecture 10. Wildberger goes through different equations for lines: functions (calculus), cartesian and parametic. His example of cutting the Mobius band and not disconnecting all the lines reminds me of a lecture from Erik Demaine's advanced datastructures class and his fascination with origami. We get a new operation dmin(), when he finds the 'determinant of minors' you can see how easy this would be to write these operations as a program, and many optimized libraries already exist for doing so we won't be doing this by hand, this is to understand what those libraries do.

50:32 you can pause the lecture and easily calculate the dmin yourself for practice, in fact whenever he does some operation like finding the determinant see if you can do it yourself first.

Lecture 11 the algebra of 3x3 matrices. A diagonal matrix, with constant diagonal, is a model or copy of the field of rational numbers, every number has a constant diagonal matrix. He goes through various transformations like reflections, projections and rotations in 3D. He shows the standard cos/sin rotation then rational trigonometry rotations, which makes rotations easier to manipulate when we are just learning, the basic rotations using sin/cos are the same. Generalized reflections and projections are exampled, you don't have to memorize everything here just know what's happening: a line can project itself or relfect itself onto the plane, see if you can figure out what the matrices are doing as they all have patterns. The perpendicular definition, now you know why ax + by + cz = 0 is the plane equation. There's only two exercises, this was a good lecture of stuff that's usually not described at all in linear algebra texts.

The marketing indicates it's curriculum is primarily targeted AMC8 to AMC12 (depending on module) middle school competition difficulty emphasizing how to solve problems without requiring memorization. The idea is that everyday for a month you do a challenge problem and then has numerous follow up lectures explaining the challenge as each problem is designed to teach several topics at once. Even though this is directed at kids the content is sophisticated as it's math competition techniques. The pre-reqs are you know everything from pre-algebra and 'are already experienced in math competitions' so things like moems.org the math olympiads for elementary and middle schools.

Singing up for free demo tier requires all kinds of age verification bs like $0.33 non refundable deposit of proof of ID for various regulatory reasons since this is a jr highschool targeted site so I will save you the step and look around.

This is his free site however the original Expii looked like this https://v1.expii.com/grandmaster but you won't for some reason find that listed anywhere on their main page, and all the practice problems are gone too. The original concept was Expii is a personal tutor to fill missing gaps in the prereqs dependency chain though since this part of the site disappeared off the main page I'm assuming it's depreciated and being replaced with Expii solve. You can still use expii of course just I don't find it as effective anymore, it's like a community contributed Khan Academy which if you're looking for a specific topic is still a great resource.

Every Friday from 4:00pm – 5:00pm EST has a live session of the same content aimed at highschool competitive algebra and geometry where you can ask him anything, for example in 07/08 Wed stream he shows how to solve \(3^{x^2 - 2xy}\) = 1 where x² = y + 3, we can do this too because we've read Tao's book but he always has some interesting optimized method being a national olympiad coach.

Here at 41:13 he answers even nonsense questions giving a tutorial on logical implication. Somebody after asks him what should students focus on breadth or depth? He answers that you get both by doing old problems from math competitions.

A list of questions asked and answered in all these live session are on the forum which you can't view without registration, and if you register you can't post unless you're in the Daily Challenge which is probably a good idea since it's all young kids you don't want to post there anyway.

This is an interesting method he found while we was working on Daily Challenge content trying to better explain quadratic equations without having to use the quadratic formula. The full version is here. Wildberger did something similar, rewriting the quadratic formula using Newton's method of approximation to make it easier for students.

Prof Loh recently put most of his PhD course 21-738 Extremal Combinatorics up on YouTube which uses similar teaching style of his live sessions. It's a theoretical course, if you've never seen one you can still follow along knowing only what we know now combinatorics has low barrier to entry because it's all finite math/sets. These courses aren't always linear they will on different days cover different theorems. Not that we expect to be able to solve hard problems in combinatorics or anything but it's worth watching to see how he plays around with structures and objects to test their properties in a theory course. This course has interesting content like Ramsey Theory where every large enough system no matter how chaotic it appears there must contain large structured subsystems. You can conjure up things with probability to show they exist, then use extremal techniques to 'derandomize' to reveal a deterministic solution.

You can get a crash course on graphs here and that same playlist has a beginner probability summary lecture good enough to follow along many of the lectures if you wanted.

This is a professional development seiminar for teachers given by Prof Loh on linear algebra specifically why is the formula for matrix multiplication constructed the way it is, and how to demystify the determinant. I audited them to see if we could use them here, the determinant lecture is definitely worth your time.

Returning back to Tao's Analysis I let's learn what a limit is after using formal limits for all of chapter 5. Reminder if you haven't seen it Wildberger has a lecture on limits. Think of any infinite convergent sequence such as 1.9₁ 1.99₂ 1.999₃ … converging to 2 and that you have taken such a sequence and stuck it in an imaginary infinite array and indexed every value. This sequence does not end and goes on forever getting closer to the least upper bound. Everytime you subtract the limit from the next sequence entry you get a new smaller epsilon if it is a converging sequence. Try this yourself subtract (1.9 - 2), (1.99 - 2), (1.999999 - 2) and take the absolute value. If you choose an arbitrary epsilon of say 1, then every entry in the remaining infinite sequence will have an equal or smaller distance from 1 if it is converging, which is what the definition means by 'every epsilon > 0'. The further you go down the infinite sequence the smaller this epsilon becomes so that 'for all positive epsilon' there is some index in this array where if you pick any arbitrary epsilon like .0000000001, every future index will contain a value equal or smaller than that epsilon.

Prop 6.1.7 notice he has assigned epsilon to the value |L-L'|/3 which is why d(L,L') \(\le\) 2epsilon is d(L,L') = 2|L-L'|/3 I somehow missed that assignment and spent a few minutes wondering what was going on. He wanted a tiny displacement but why divide by 3 instead of 2? I'm guessing to set up the triangle inequality.

In Remark 6.1.9 we throw out the starting index, I also noticed he changed the index type from naturals to integers in these definitions. Try some exercises:

We patch the definition of real numbers with negative and positive infinity elements that are distinct from all other real numbers. All we can do with these two new objects is equate things to them, something is infinite if it equals negative or positive infinity. Tao points out cancellation of infinity elements leads to nonsense conclusions.

Example 6.2.7 and def 6.2.6 if a set 'E' contains negative infinity, sup(E) is that set with negative infinity removed and the least upper bound taken, so his example E = {-1, -2, ….} U {negative infinity} results in the set {-1, -2, …, -\(\infty\)} so there is a set of infinite descending numbers unioned with an object 'negative infinity' and sup(E) is -1 because we remove -\(\infty\), but are still left with the first set of {-1, -2,..} even though it technically is also negative infinity it is not the object \(-\infty\). Example 6.2.9 that set is equal to infinity but does not contain the actual element infinity.

Example 6.3.4 the supremum = 1 is obvious but how do we prove it? The sequence is a_n \(\le\) 1 for all n \(\ge\) 1, so any element x of this set x < 1, but since a₁ = 1 and is in the set, we have x < a₁ so x is not an upper bound, and 1 is the upper bound. Similar reasoning for the infimum, a_n > 0 for all n \(\ge\) 1, so we have x > 0 for all elements in this set. By 5.4.13 Archimedean property we have a positive integer N such that Nx > 1 or 1/N < x, so if 1/n < x that means x > 0 is not the lower bound of this sequence since a_n < x so we must have 0 as the lower bound. There is a completely different solution here you may want to look at, using 1/n \(\le\) M for all n > 0, so for n = 1 we have 1/1 \(\le\) M or 1 \(\le\) M for the supremum. The infimum is derived using a contradiction. It doesn't matter how you prove it.

Prop 6.3.10 why is it decreasing? Because \(a_{n+1} \le a_n\), pick any arbitrary x between 0 and 1, like 1/2. a₁ is (1/2)¹ or 1/2. a₂ is (1/2)² or 1/4. a₃ is (1/2)³ or 1/2 x 1/2 x 1/2 or 1/8. This fails when x > 1 since it is no longer a decreasing series and is divergent. Tao asks to look at the first chapter again, which we skipped but now can read. The problem with 1.2.3 is that it assumes there is a limit for all real numbers xⁿ, a limit can only exist if 0 < x < 1 for xⁿ as n goes to infinity.

You can look up a proof of the monotone sequence theorem here. If you forgot all the inequality manipulations he did here review Tao's chapter on them again.

If this chapter doesn't make sense find a real analysis playlist. Since this is an analysis book we are exploring metric space (the real line) and now getting into topology territory. Example 6.4.4 we find out limits are just a special case of limit points.

Def 6.4.6 note the new notation is "the supremum of all the elements from a_N onwards" so the example 6.4.7, what's the supremum of a₁ and all future indices? 1.1. What's the supremum of a₂ and everything past it? 1.001. What's the supremum of a₃ and everything past it? 1.001 again. You are looking ahead not before the current sequence index.

Prop 6.4.12 L⁺ and L^- are extended real numbers so we can equate them to infinity. In the same proposition a) and b) seem to be saying that there exists an infinite amount of real numbers in the sequence bigger or less to wherever we happen to be when we take the limit superior or limit inferior. For c), the infimum of the entire sequence will be less or equal to the limit inferior because the limit inferior doesn't look at all the previous sequence entries, same with the limit superior so it too will be less or equal to the supremum.

Squeeze test introduced here in 6.4.14, and Tao finally completes the real number definition.

Very short chapter, the proof of 6.5.1 is a standard point-at-definitions style proof, see Lemma 5.6.6(a) and (b) and (g), if L = 1/x^1/k then L^k = (1/x^1\k)^k\1 which is L^k = 1/x. For the exercise with x^q, look at Def 5.6.7 and 5.6.9 again.

Example 6.6.2, a_2n is multiplying the indices by 2, producing the subsequence a₀, a₂, a₄ like in his example. It just means if you apply a function to the indices that is strictly increasing, you get another sequence and can call this new sequence b_n or c_n. Theorem 6.6.8 proof: since we already declared a bounded sequence meaning a_n \(\le\) M for all n in N, then (negative bound) \(\le\) a_n \(\le\) (positive bound), so this can't be equated to infinity as it lies in between these two bounds. This means L is a limit point of this sequence and within this sequence one or more subsequences exist that converge to L. Tao teaches us some topology terms in 6.6.9.

I didn't do any exercises here, later when we do probability and the Putnam book we will revisit all these definitions again as needed all you need to know that if you apply an increasing function to the indices of an infinite sequence you can probably find a subsequence in there somewhere that has a limit point.

Here we learn what things like x^e or x^π mean. TODO

We finally arrive at pre-calculus. TODO

This is an introductory recorded Putnam seminar for CMU students. Since CMU is closed, this seminar is being streamed on Youtube.

The book Putnam and Beyond is recommended by Prof Loh in his Putnam seminar, written by one of his IMO coaches. It contains mainly competition level problems, showing us how to solve something then asking the reader to try it themselves. Every problem in the book has a worked out solution with the most interesting or elegant proof they could demonstrate to teach the reader different strategies. Prof Loh says we should spend about 2 hours trying to figure out a problem and increase the time we spend as we go, until we build up the stamina needed to focus on a problem for many hours (this can be split up, not doing it all at once). We can also go through this lecture series on doing research math aka 'methods to solve any problem by hook or by crook' as we go through Putnam and Beyond, and learn things like Proof Theory and natural deduction when they come up.

Let's look at the second edition of Putnam and Beyond by Andreescu and Gelca and see how far we get, alternating between Tao's text and Wildberger's Linear Algebra course we're already doing. Remember this workshop is just to generally learn undergrad math, all the specific AI needed math will be done in that workshop. This is a hard book but will get easier the more proofs we see.

Coprime if you look it up means two numbers where their highest common denominator is 1. The indices for N = p₁ p₂.. p_n means multiplication (factorial), as there is no commas between them so you have a list of primes p₁, p₂, .. p_n and assume p_n is your whole list of finite primes. Then multiply every index together and add 1, that will still be divisible by some prime p yet it's not in our original list of primes. Try the example, suppose p_n is 4 primes, so 2, 3, 5, 7, and p_n! + 1 is 2x3x5x7 +1 or 211. That is divisible by 11, another prime but it's not in our list, hence Euclid claiming there will always be more primes. The second example of this proof, 'repunit' means repeated units like 11, 111.. and try the examples yourself of 10x_n! + 1 so take x₁, multiply it by 10, add 1, you get 11, then you get 111, etc.

The Euler polynomial example, recall that polynomials are written in descending powers of n, so a₂x² + a₁x¹ + a₀.

todo after we finish Tao's Analysis I since we need Calculus.