The most straight-forward way of showing something is invertible is simply find an inverse element and check that it is indeed the inverse. I think there is a very good candidate for the inverse in this case.
As a rising junior in high school, I think you're in a good shape in terms of determining your major. I think it's important to pick a "direction" before you enter college (e.g. not specific enough to claim a major, but something along the line of "science," "humanity," "social science," or...
A good point. One more thing to add to my previous comment---Calculus I is a very sequential course, meaning, the material you learn in the course will build up what you have learned previously. It's like building an apartment complex with multiple stories---you first build the ground floor...
I think the hardest part of the book is actually the second chapter, which is on metric spaces. I would do my best to avoid metric spaces and basic topology from that book for the first time, especially about the compact sets. As soon as the definition of a compact set is given (in terms of open...
First, at least give your best shot for the final exam---some instructors like to bump up a grade of someone who showed some improvement on the final ("if you can do well on a cumulative exam, perhaps the student should deserve more than a failing grade"). Of course, not every instructor has...
Also, make sure what class those reviews are coming from. I know one good prof with 20 low ratings simply because there was one semester he had to teach a business calculus course with 500 students in it.
If you are not sure, then you should probably take a little bit of both, and that "little bit" depends on how flexible your schedule/program is (e.g. some math departments can let you take whatever you want and still graduate, whereas the others want to take you specific number of courses from...
Thanks! Based on the course descriptions, Analysis 2 sounds like a very reasonable sequence that follows Analysis 1. It can hard in a sense that the materials build upon what you learned in analysis 1 (afterall, you need to be solid on the analysis of R^1 in order to learn the analysis of R^n)...
I'd say Kosniowski is more elementary than Massey or Munkres. It is brief, and cover less material than the other two, but it does have a pretty good exposition of the subject, with a good balance of abstract idea and concrete examples.
Massey should be good if you already familiear with some...
Getting any PhD from a decent math program, including MIT, is challenging. I don't think this is something that we can easily compare between two schools (as definition of "harder" differs from a person to another person).
Getting INTO a PhD program in mathematics at MIT, on the other hand, is...
I think so too :wink: I now realize that I just happened to be a bad neighborhood where my point happened to be a maximum, and it took some work to walk away from that neighborhood.
(Of course, local max can be a global max, but most people probably won't reach there anyway...)
I am having a difficult time remembering what else was there besides what you have already mentioned in basic trigonometry. Certainly, there are some other topics like inverse trig functions, sec/csc/cot, and some application problems (angular velocity, etc), but they were just applications of...
I agree with micromass. You will hit a wall or sometimes, or even a ceiling. One time when I got frustrated with my real analysis course, and I asked my adviser whether I hit the maximum of my mathematical potential. And he answered "yes, but more like a local maximum." (BTW, I did not quit math...
You might want to let us know that...
1) Whether you are learning linear algebra for the first time or not.
2) Whether you are interested in the theory or applications of linear algebra.
There are different linear algebra books for different purposes, so clarifying this matter would help us...
I must apologize that I won't be answering your question.
If you already know the materials in Spivak, then I would pick any of these books, and read only the chapters that deal with applications. For example, Stewart has a chapter on optimization problems using the theory differentiation...
For Hilbert spaces, I strongly recommend Young's https://www.amazon.com/gp/product/0521337178/?tag=pfamazon01-20&tag=pfamazon01-20. I thought the book was very easy to read, and believe it is wonderful introduction to the subject.
In most of my math classes, B+ usually means "You did well for the most part, but you still have some room for improvement." This is not a bad grade, but it does suggest that you have a weakness. Some people think learning math is kind of like being a "tough guy", and you are not supposed to...
I feel a little sympathetic to this. When I took a topology course during last fall, I thought I was doing everything right; I did all the homework problems as well as most of the extra credit problems, did well on the midterm, and I thought I had a pretty good grasp on the material until I...
I used Czes Kosniowski's https://www.amazon.com/dp/0521298644/?tag=pfamazon01-20&tag=pfamazon01-20 for my topology class. The first ten chapters of this book are on point-set topology that is suited for anybody who is learning topology for the first time. I thought the book was very well...
Hi,
I will be starting my graduate study in mathematics in August 2011, and I was thinking of reviewing mathematics that I have learned while I was an undergraduate as soon as my summer vacation begins in June.
The topics that I am considering to review include algebra (abstract and...
I'm not a physics major, so my comment may not be that helpful.
I don't think retaking a class to improve your grade would hurt your chance of getting into a graduate school, unless your original grade for that class is like B or above. If you feel like you haven't learned the material properly...
What is your subject field? If you are planning to go to a grad school in math, physics, or possibly chemistry, your general GRE scores don't matter that much as long as the rest of your application looks good; general GRE is more of a formality in these subjects.
Different story for the...
Unless you're in an applied math program; yes, upper division math courses are very proof oriented.
I think the types of proofs you do in discrete math courses are more foundational than the ones you do in upper division math courses (at least that seems like it at my school). You learn...
I am a math major who have taken plenty of high level math courses, but calculator becomes absolutely useless once I finished taking lower-division math courses (e.g. calculus, diff eq, elementary linear algebra). This is because
1) Most of the high-level math courses are proof-oriented, so...
See here:
http://math.scu.edu/~eschaefe/gre.html
On this website, there is a list of the number of problems (from the four sample exams) that are from the "Additional Topics" sections in the math GRE test. It turns out that there are more probability questions than any other topics in the...
Five paragraph essay is a format of an essay that is most commonly taught in the United States. I suggest you to do a little Google searching, but here are my short explanation and suggestion.
As the name suggests, it is an essay that is built from five paragraph. The first paragraph is...