Most Popular
1500 questions
114
votes
3 answers
Why are infinitely dimensional vector spaces not isomorphic to their duals?
Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$).
I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the vector space $V=\mathbb F^{(\kappa)}$ (that is an…
Asaf Karagila
- 393,674
114
votes
7 answers
Why is it important for a matrix to be square?
I am currently trying to self-study linear algebra. I've noticed that a lot of the definitions for terms (like eigenvectors, characteristic polynomials, determinants, and so on) require a square matrix instead of just any real-valued matrix. For…
Beneschan
- 1,083
- 2
- 9
- 10
114
votes
8 answers
Much less than, what does that mean?
What exactly does $\ll$ mean?
I am familiar that this symbol means much less than.
...but what exactly does "much less than" mean? (Or the corollary, $\gg$)
On Wikipedia, the example they use is that $1\ll 9999999999$
But my thought on that is that…
user285523
114
votes
4 answers
Why do we define quotient groups for normal subgroups only?
Let $G \in \mathbf{Grp}$, $H \leq G$, $G/H := \lbrace gH: g \in G \rbrace$. We can then introduce group operation on $G/H$ as $(xH)*(yH) := (x*y)H$, so that $G/H$ becomes a quotient group when $H$ is a normal subgroup.
But why do we only work with…
Aleksei Averchenko
- 8,023
114
votes
3 answers
When is the closure of an open ball equal to the closed ball?
It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal
to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set and define a…
Alex Lapanowski
- 2,926
- 2
- 20
- 23
113
votes
2 answers
Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square
Question:
For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions.
My try: It is clear that the function
$$f(x)=x$$ satisfies the given conditions, since:
…
math110
- 93,304
113
votes
9 answers
Is $[0,1]$ a countable disjoint union of closed sets?
Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
Kevin Ventullo
- 3,692
113
votes
16 answers
If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum_{n=1}^\infty a_n<\infty$, then $\lim\limits_{n\to\infty}{n a_n} = 0$
I'm studying for qualifying exams and ran into this problem.
Show that if $\{a_n\}$ is a nonincreasing sequence of positive real
numbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \rightarrow \infty} n a_n = 0$.
Using the definition of…
dls
- 4,636
113
votes
14 answers
Expected Number of Coin Tosses to Get Five Consecutive Heads
A fair coin is tossed repeatedly until 5 consecutive heads occurs.
What is the expected number of coin tosses?
leava_sinus
- 1,243
- 4
- 10
- 5
113
votes
9 answers
Why should I "believe in" weak solutions to PDEs?
This is a sort of soft-question to which I can't find any satisfactory answer. At heart, I feel I have some need for a robust and well-motivated formalism in mathematics, and my work in geometry requires me to learn some analysis, and so I am…
A. Thomas Yerger
- 17,862
- 4
- 42
- 85
113
votes
7 answers
Is the vector cross product only defined for 3D?
Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as
$$
\vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n
$$
It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and $\vec b$,…
VF1
- 1,993
113
votes
5 answers
Getting Students to Not Fear Confusion
I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro to Real Analysis.
Since this was essentially…
Matt
- 7,358
113
votes
19 answers
Why is the volume of a sphere $\frac{4}{3}\pi r^3$?
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for…
Larry Wang
- 9,513
112
votes
3 answers
Expectation of the maximum of gaussian random variables
Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large?
If $F$ is the cumulative distribution function…
Chris Taylor
- 28,955
112
votes
24 answers
Why do we still do symbolic math?
I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one.
Numerical computations, to my understanding, never deal with irrational numbers, but only rational…
totalnoob
- 1,243