In complex analysis |z| is called the the modulus. Since the complex field is not well ordered (meaning we can't say stuff like 3<5) then we cannot generalize the real value absolute value into the complex field. Instead we define |z| = Sqrt[x^2 + y^2]. This is consistent with the real valued...
Oh this stuff happens. I once had probability greater than one and didn't notice it until 2 weeks later when a friend of mine says, it all makes sense until here (the 5th equation i wrote.) Sometimes you have lapse in judgement. Get up start again and hope for the best.
Be careful with this attitude. Doing work you don't want to do is important so that you may do the work you want too. There are to many stories of kids who were really bright but never worked hard so when they hit University they just fumbled around after a while. I suggest along with working...
So that is 2 math classes per semester for your sophomore year? Sounds about right. As for if you should take complex analysis before or after topology, that really depends. Each school teaches that course differently. At my school, it depended on real analysis. If people at your school say...
1) presence of researchers working in a subject of interest.?
This was the most important factor for me. A strong research group had to be present with well known people in the field.
2) supportive grad program
This was also key. I wanted to feel that the program was behind me and that I...
I did math at the UT- Austin, so i'm bias. Out of the three schools listed, which one would benefit you the most is a tough question to answer. It really depends on what you value more, and what you expect. I choose UT over Princeton for two reasons: 1)I was a big fan of Austin 2)I hate...
Math GRE is over hyped. I tend to agree with mathwonk's assessment of the test . It is essentially a cutoff point.
mrb if you feel you will one day be able to dominate the GRE than you should relax a bit. I believe Mathwonk was simply saying that if you can't do well on the GRE then you...
heh....you don't. You take it all, you learn it, you say how much it sucks but by the end of the semester you know some cool stuff and it makes it all worth wild.
Really, it's just about reading the book, asking questions in lecture, and doing the homework. When you learn these theorems, you need to sit there and think about what it means and if you really want to get ahead think about what more theorems the the theory or definition implies. It's...
Copying examples is fine for lower division math classes, but as you move up to more abstract classes, then it becomes little use. It's more helpful in the long run to learn definitions, learn theorems and see why they are true, and then apply those theorems and definition to problems. The...
Yep, let's break this down into parts:
1)Assume Ax = b has a solution, then b can be written as a linear combination with the vectors from the columns of A. Since the span is the linear combination of the vectors a1 a2 a3 ... an then b is in the span. So then use your definition.
2)Now work...