My first month in Math.

If you find the solutions PDF to your textbook, you don't have to do all the homework.

Do the exercises. If allowed, discuss them with a few buddies, but make sure you write them down yourself and see to it that you understand every step for yourself. This is the most important part. Go to the office hours and ask questions. How to ask questions: Make sure to be specific and have tried at least something ("not knowing what to do in general" is not a serious question). Also, know your definitions or look them up. Most questions come from not knowing exactly what a certain object/thing is.
If there's a book/script for your lecture around, prepare the next lecture - just go through the next few pages and check out what will be done, it will help you understand/categorize what's happening in the actual lecture. This obviously includes attending the lecture. Go through the lecture again afterwards.

At this level it is best to learn through solving problems rather than reading first. Start with the simple ones and work your way up, it will make sure you understand your theory instead of just knowing it. Also set theory is a pretty good foundation for analysis (well for anything really), so make sure you got it down good. Other than that just ask away at your lecturer, I'm sure they're happy that someone is interested in their stuff (I assume you're starting university?)

On September 28 2013 18:15 corumjhaelen wrote:
You should do the exact opposite. Solving excercises is a waste of time if you don't know everything in your lesson perfectly, including proofs, examples and so on. If you don't like your textbook, go find another one you like more.

On September 28 2013 18:41 corumjhaelen wrote:
Can't help you for the textbook, because I had very competent teacher in undergraduate (classes prépa), and most of those I use today are in French... Just go to your unis library, look at a few of them, and choose one you like, we all have different taste anyway.
I'm going to repeat myself. You're doing it wrong. It's no use doing any exercise in linear algebra if you don't know exactly what linear independance is. The best application of the concept is understanding what finite dimension is. It is (or will be soon) in your lessons. Basic examples help to, but basic is the important word here.
You have to learn perfectly what's in your lessons. Or it's like you're trying to build the roof before the foundations of your house.
If you do that well, in a few months, you'll get used to the pace, you'll be able to learn the curriculum faster, and then you'll find time for more exercises. Anyway most ideas you need to solve exercises are... in your lessons.

A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 5 miles north and 6 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface). The person might add, "The place is 7.81 miles northeast of here." Although this last statement is (approximately) true, it is not necessary.
In this example the "5 miles north" vector and the "6 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "7.81 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary.
Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, n linearly independent vectors are required to describe any location in n-dimensional space

On September 28 2013 18:41 corumjhaelen wrote:
Can't help you for the textbook, because I had very competent teachers in undergraduate (classes prépa), and most of those I use today are in French... Just go to your unis library, look at a few of them, and choose one you like, we all have different taste anyway.
I'm going to repeat myself. You're doing it wrong. It's no use doing any exercise in linear algebra if you don't know exactly what linear independance is. The best application of the concept is understanding what finite dimension is. It is (or will be soon) in your lessons. Basic examples help to, but basic is the important word here.
You have to learn perfectly what's in your lessons. Or it's like you're trying to build the roof before the foundations of your house.
If you do that well, in a few months, you'll get used to the pace, you'll be able to learn the curriculum faster, and then you'll find time for more exercises. Anyway most ideas you need to solve exercises are... in your lessons.

On September 28 2013 19:45 Toadesstern wrote:
Tip 1) Get a study group. Especially in maths that's really helpful imo. There's just situation where you have to discuss and work on things in a group. Make sure you're all on the same level. Doesn't help anyone if one of them explains EVERYTHING to the other guys all the time. 3 to 4 people and you're good.

Tip 2) Start doing your homework early on and don't do it the last day before you have to hand it in. You'll have situation where you just won't get anywhere and need a day or maybe two off due to some kind of brainblockage. You need some kind of buffer for that reason

Tip 3) Learn to live with frustration. It will happen to you that you're sitting on something for weeks, you suddenly realize you can prove everything to be wrong with a simple example and you have to start from scratch. The hardest part about math isn't math, but being okay with those things imo.

On September 28 2013 19:45 Toadesstern wrote:
Tip 4) Make sure to get your Ana 1 done properly. That's what people have problems with. LA 1 and set theory is super easy. LA and set theory is easily doable on your own. Ana 1 will probably cause you troubles if you don't have some discussions with other people because you're doing everything on your own.

On September 28 2013 20:06 Plexa wrote:

As a PhD student in math and having been a TA for many years, this, this, a thousand times this.

The #1 reason why people get stuck in math is because they don't know their definitions. (The #1 error, however, is basic arithmetic! haha).

Whenever you're posed an exercise or problem, you need to be thinking about what the assumptions are that are given in the problem. What do those assumptions mean? i.e. what is the definition of those assumptions. For instance, you could be given a question like suppose that U and V are subspace are R^3, show that the union of U and V is a subspace. To show this obviously you need to us the fact that U and V are subspaces, and to do that you need to know what a subspace is! While this was a (trivial) abstract problem, even something like show that the union of U=span{(1,0,3)} and V=span{(3,4,5)} is a subspace requires the definition to solve!

In time, you'll build up from definition > application to definition > theorems > application!

On September 28 2013 21:03 Toadesstern wrote:
oh another thing, though probably not what you're asking for:

Just like with us men, the percentage of chicks looking decent and being normal is a lot better if you're looking at those who're doing maths to become a teacher. At least here in Germany they have to do the same Ana 1-3 and LA 1-3 and not some kind of "math for biologists" (or whatever it'd be for teachers) so you'll be sitting in the same lectures.

The ones doing math to get a bachelor and plan on making a masters afterwards are a lot more nerdy, which I probably don't have to explain to anyone here. So if you're looking for people who are a bit more down to earth you should be looking over there. People starting out in maths, when comming from highschool tend to be huge dicks the first 1 or 2 semesters until they realize they aren't geniuses.

On September 29 2013 04:59 Hryul wrote:
This confuses me. Our math assignments were really hard and it took regularly multiple days to solve them. The exams were "rather" easy in comparison, mostly due to time limits. so you should take that with a grain of salt.

also: why hasn't nobody pointed out that plexa's example is wrong? the union of subspaces is, in general, not closed under vector addition.

On September 29 2013 05:23 Geiko wrote:


I think he meant the sum instead of the union

i also think he is wrong in how you learn math. If you try to learn all your definitions by heart before actually knowing what you're talking about, you're just setting yourself up to be a monkey savant, and you'll never be any good.

Proper math should be learned through trial and error.
1)Read your lessons quickly
2)Try to do some exercises.
3)Go read specific chapters in the book to help you when you are stuck.
4)Do more exercices.
5)Repeat steps 3) and 4)
5)Once you have a good feeling (intuition) for what the objects means and how they work, go back and learn your lessons by heart.

If you try to learn your lesson by heart without first knowing what it means, you'll spend hours not having fun, and chances are, you'll get decent grades when it comes to solving direct applications of theroems but still have no clue what you are doing.

On September 29 2013 06:05 Geiko wrote:
Definitions are made to be as concise and precise as possible. Most of todays axioms, definitions and theorems were written way after they were actually invented and used.

For instance the definition of a line (AB) in a metric space (E,d) is
{P€E / d(A,B)=|d(A,P)±d(P,B)|}.
If you don't have a intuitive knowledge of what a line truly is, you're never going to able to use this definition other than in simple problems.

You have to have a good feeling of what mathematical objects are before learning your definitions.

On September 29 2013 06:19 corumjhaelen wrote:
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise.

On September 29 2013 07:44 Geiko wrote:


That's what I'm saying You get intuitive knowledge of mathematical objects by applying definitions to simple problems. But you never truly understand their purpose until you've figured out by yourself how they work and why they are useful.

Telling people to work on their lessons very hard before doing any problems is like giving them a book full of problems with the solutions written right next to the problems. Most people will understand the solution, and those same people will immediately forget how to do it next time they see it. If you spend time on problems you can't solve with your current knowledge, you understand why you need the elements in your lessons.

In hindsight it's easy to think that the best way to learn math is to learn definitions, then apply them intelligently, because that's what you do in math once you've understood. But the people who advocate such a way of teaching forget that this is not at all how they themselves learned math. They forget that they were deriving and integrating before knowing what it meant, that they were working in finite commutative rings before knowing what those were etc...

On September 29 2013 07:51 corumjhaelen wrote:

What I mean by learning a lesson seems to be different than what you mean. The idea is not to read the course over and over, the idea is to take a blank paper, copy a few headlines, and being able to rewrite your course from A to Z, in your own way, not by heart...
Of course that's impractical in a master's degree, but I think there's nothing more important to do in undergraduate classes.
Edit : and that's how I worked in practice you know...
Edit 2 : what is a finite commutative ring I worked in ?
Every maths concept is not as intuitive as you make it sound, plus intuition is such a good recipe for disaster in so many cases, I'm a bit dubious at what you say.

On September 29 2013 08:06 Geiko wrote:


I'm sure you worked with modulus (11≡3[8]) before knowing what Z/8Z (finite commutative ring) was.

Also if you have no intuition of a math concept, then you haven't understood it. It's that simple. You might know how to solve a couple of problems with it. You might know how to use it in some circumstances, but you don't know what it is. Intuitionis what you need to be good in math and by that I don't mean get Master degrees or PHDs or even good grades in high school. Then once you have intuition, to be truly brilliant you need to be able to write flawless proofs. Not the other way around.

The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.

On September 29 2013 05:23 Geiko wrote:


I think he meant the sum instead of the union

On September 29 2013 08:06 Geiko wrote:


I'm sure you worked with modulus (11≡3[8]) before knowing what Z/8Z (finite commutative ring) was.

Also if you have no intuition of a math concept, then you haven't understood it. It's that simple. You might know how to solve a couple of problems with it. You might know how to use it in some circumstances, but you don't know what it is. Intuitionis what you need to be good in math and by that I don't mean get Master degrees or PHDs or even good grades in high school. Then once you have intuition, to be truly brilliant you need to be able to write flawless proofs. Not the other way around.