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1500 questions
70
votes
5 answers
Is the rank of a matrix the same of its transpose? If yes, how can I prove it?
I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
"The rank of a matrix A is the number
of non-zero rows in the reduced
row-echelon form of A".
The…
Vivi
- 1,439
70
votes
1 answer
Are there infinitely many "super-palindromes"?
Let me first explain what I call a "super-palindrome":
Consider the number $99999999$. That number is obviously a palindrome.
${}{}{}{}$
The largest prime factor of $99999999$ is $137$. If you divide $99999999$ by $137$, you get $729927$. This…
celtschk
- 43,384
70
votes
6 answers
Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$
I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how…
JSCB
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- 123
70
votes
3 answers
Characterizing units in polynomial rings
I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.
Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if…
anonymous
70
votes
10 answers
Why is the tensor product important when we already have direct and semidirect products?
Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do we need Tensor Product?
The Definition of Tensor…
anonymous
70
votes
9 answers
Why is $\pi $ equal to $3.14159...$?
Wait before you dismiss this as a crank question :)
A friend of mine teaches school kids, and the book she uses states something to the following effect:
If you divide the circumference of any circle by its diameter, you get the same number, and…
gphilip
- 827
70
votes
14 answers
Are proofs by contradiction really logical?
Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.
My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?
That, to me, seems the…
Simp
- 715
70
votes
6 answers
Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only "unlikely" or that the probability approaches zero or…
regularmike
- 713
70
votes
13 answers
What is the definition of a set?
From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition.
My question is what the definition of a set is?
I have…
John Doe
- 3,233
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- 88
70
votes
9 answers
Finding $\int x^xdx$
I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:
Let $u=x^x$.
$$\begin{align}
\int x^x \, dx&=\int u \, du\\[6pt]
&=\frac{u^2}{2}\\[6pt]
&=\dfrac{\left(x^x\right)^2}{2}\\[6pt]
&=\frac{x^{2x}}{2}
\end{align}$$
But…
Garmen1778
- 2,338
70
votes
14 answers
Pseudo Proofs that are intuitively reasonable
What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples of what I have in mind:
$s=\sum_{i=0}^\infty…
Michael Greinecker
- 32,841
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- 80
- 137
70
votes
4 answers
Conjugate subgroup strictly contained in the initial subgroup?
Let $G$ be a group, $H\subseteq G$ a subgroup and $a\in G$ an element of the group. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$?
If $H$ has finite index or finite order, this is not possible.
Sasha
- 1,113
69
votes
3 answers
Discontinuous linear functional
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$…
FPP
- 2,103
69
votes
9 answers
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$
I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$…
user365
69
votes
4 answers
Does the string of prime numbers contain all natural numbers?
Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
Buddha
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