I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matr...

There are so many available bases. Why is the strange number $e$ preferred over all else? Of course one could integrate $\frac{1}x$ and see this. But is t...

Calculate $\arcsin (1 + i),\> \arctan (1 - i), \>\arcsin (i),\> \arctan (2i)$. As $arcsin (z) = -ilog [iz + (1 - z^2)^{1/2}]$, then For $ arcsin(...

I have a question about parallel lines and coincident lines. According to wikipedia a parallel line is: Two lines in a plane that do not intersect or touc...

In set theory there are many types of cardinal numbers. One of them is the weakly compact cardinal. Wikipedia states the following about it: Formally, a c...

In general how to solve this equation:(with Hensel's lemma and without it) $$2x^3+7x-4\equiv0 \pmod{25} $$

Find the coordinate vector of $1+x+x^{2}+x^{3}$ relative to the following basis of $P_3$: $$B=\{1-x,1+x,x^{2}-x^{3},x^{2}+x^{3}\}$$ I have never to deal w...

The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fit...

Given the points $A(2,0), B(1,-1), C(1,0)$ and $D(0,-1)$ in $\mathbb{R}^2$, using Green's theorem I have to calculate the following integral: $$\int_...

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a s...