The process of studying mathematics without formal instruction. _Don't_ use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when the fact that you're self-studying is what your question is _about_.

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A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
23
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3answers
2k views

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Princicples of Mathematical Analysis," or "Baby Rudin," ...
29
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8answers
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Good abstract algebra books for self study

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however,...
29
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9answers
61k views

Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
57
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4answers
7k views

Do you prove all theorems whilst studying?

When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
7
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4answers
2k views

Expanding problem solving skill

I have a great passion for Math but my lack in problem solving skill always keeps me away from the "good stuff". I always wanted to be better at Math and one of the things I figured out was to keep ...
56
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9answers
9k views

How to effectively study math?

Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble ...
78
votes
6answers
17k views

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the ...
233
votes
7answers
17k views

Best Sets of Lecture Notes and Articles

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
284
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21answers
16k views

On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole". For example, suppose you come across the novel term vector space, and want to learn more ...
20
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2answers
6k views

Path to Basics in Algebraic Geometry from HS Algebra and Calculus?

In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ...
14
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1answer
7k views

How to self study Linear Algebra

I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is ...
10
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2answers
372 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
4
votes
2answers
2k views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
6
votes
4answers
2k views

Can a function with just one point in its domain be continuous?

For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$. I have the next context: 1) According to the definition given in Spivak's book and also in wikipedia, since $\...
197
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28answers
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Too old to start math [closed]

I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some ...
19
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10answers
7k views

A Book for abstract Algebra

I am self learning abstract algebra. I am using the book Algebra by Serge Lang. The book has different definitions for some algebraic structures. (For example, according to that book rings are defined ...
15
votes
2answers
5k views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
4
votes
1answer
148 views

Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$

$a,b,c$ are positive reals with $abc = 1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$$ I try to use AM $\ge$ HM. $$\frac{\dfrac{1}{a^3(b+c)}+\dfrac{1}{b^3(...
6
votes
2answers
494 views

General question on relation between infinite series and complex numbers

This is a strictly preliminary question. I hope to elicit some discussion/s which will lead to a more acceptable form for the question on this site. I'm trying to understand how the study of the ...
1
vote
0answers
100 views

Finding total number of multi-sets

I am provided with a multi-set, let's say S with elements as [num1, num2, num3] and these elements are integers (both negative as well as non negative). As this is a multi-set, elements in the multi-...
32
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8answers
14k views

Game theory - self study

I want to self study game theory. Which math-related qualifications should I have? And can you recommend any books? Where do I have to begin?
19
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6answers
3k views

Book on the Rigorous Foundations of Mathematics- Logic and Set Theory

I am asking for a book that develops the foundations of mathematics, up to the basic analysis (functions, real numbers etc.) in a very rigorous way, similar to Hilbert's program. Having read this ...
21
votes
1answer
2k views

Learning path to the proof of the Weil Conjectures and étale topology

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
10
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4answers
4k views

Preparing For University and Advanced Mathematics

I am currently a college student, computer programming, who has developed an intense passion for mathematics. Following my graduation I wish to pursue a University degree in mathematics, the September ...
18
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4answers
2k views

Grasping mathematics

First, I'm not trying to make this sound like a "poor-me" story. I understand fully that every decision I've made leading to this is my fault. I am genuinely looking for advice. So, I am a high ...
8
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3answers
2k views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
6
votes
2answers
284 views

What all maths do I need to know to become good at machine learning.

I am a computer science engineer and I took a couple of maths classes in my first year they were on Fourier series(not transform) partial differential equations, vector calculus, infinite series ...
5
votes
7answers
3k views

Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...
2
votes
3answers
480 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? [duplicate]

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for $a,b&...
1
vote
4answers
153 views

If $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$

Given $a_n$ be a sequence and IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ I do not know how to do this. Can someone help me with this? Thanks ...
3
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2answers
769 views

Generating sequences using the linear congruential generator

I came across the linear congruential generator on Wikipedia: http://en.wikipedia.org/wiki/Linear_congruential_generator I gather that for a particular choice of the modulus, multiplier and ...
3
votes
2answers
186 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
2
votes
1answer
166 views

Help proving exercise on sequences in Bartle's Elements

Self learning Analysis and found the following exercise in Bartle's Elements of Real Analysis: Let $X = (x_n)$ be a sequence of strictly positive real numbers such that $\lim \left({\frac {x_{n +...
48
votes
2answers
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What is the proper way to study (more advanced) math?

Here's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I think ...
81
votes
16answers
11k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...
37
votes
9answers
5k views

Is complex analysis more “real” than real analysis?

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
17
votes
2answers
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Relearning from the basics to Calculus and beyond.

Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ...
9
votes
4answers
3k views

Books, Video lectures, other resources to Teach Yourself Analysis

So my limited mathematics education has been especially ignorant of analysis. In this vein, I'd like to teach myself some of the introductory basics. I'm intrigued by sources that might contain ...
10
votes
3answers
3k views

Preparing for Mathematics Olympiad

I am preparing for Mathematics Olympiad , can any one suggest me some books to prepare for olympiad ? The topics that usually come up involve: congruence modulo $n$, inequalities , number system, ...
6
votes
2answers
2k views

Strategy to improve own knowledge in certain topics?

My background is physicist/engineer, and at university we studied only particular topics in mathematics. Some of the courses were never used and faded away with time, and I need to refresh the ...
11
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5answers
7k views

Multivariable Calculus books similar to “Advanced Calculus of Several Variables” by C.H. Edwards

I am currently trying to teach myself multivariable calculus using C.H. Edwards' "Advanced Calculus of Several Variables", but the text unfortunately doesn't have very many problems with solutions. I'...
8
votes
7answers
1k views

Good Number Theory books to start with?

I'm in Grade 11. I'm interested in elementary number theory and would like properly study it. I'm not intending to enter any competitions.
7
votes
9answers
5k views

Good Textbooks for Real Analysis and Topology.

I'm currently in my 3rd year of my undergrad in Mathematics and moving onto my 4th year next year. I took a course in Real Analysis I, but the professor was very confusing and we didn't use a textbook ...
12
votes
1answer
525 views

How to Self-Study Mathematical Methods?

Edit: Ok, user Chinny84 made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods. Thanks to his recommendation I ...
10
votes
2answers
5k views

Rigorous Text in Multivariable Calculus and Linear Algebra

So I'm wanting a solid math book for Christmas. I have a solid background in Calculus and am currently working through baby Rudin. I really want a rigorous book dealing with multivariable calculus ...
9
votes
2answers
556 views

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
9
votes
3answers
433 views

Beginning of Romance

I am a 17-year old student in India, in the standard 12th grade. Recently, I found the fascination in mathematics, and I am eager to dig in further. Currently, the only textbooks I have are the ones ...
5
votes
1answer
2k views

Looking for a logically coherent book for the self-study of differential equations

I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify. By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc. By ...
6
votes
5answers
919 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...