Given the following rotation matrix $$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{m...

Prove that $f: \mathbb{N} \to \mathbb{Z} $ is a bijection where the relation between $\mathbb{N}$ and $\mathbb{Z}$ is as follows: $f(n)= \frac n2$ if n is...

I'm having trouble with questions like these. In the first image, the original function is what is the two sharp lines and a semicircle in between. I...

$\left [ \left \{ 1,2,3,4 \right \}! \right]^{-1}$Is this the correct notation if one wanted to obtain the factorial for each number in a sequence and the...

I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question ...

What is the general formula for a diagonal parabola facing a given direction with a given vertex (x,y)?

I have been given a wave function and been tasked with verifying it has been normalized. $$\psi(x, 0) = \left(\frac{2\alpha}{\pi}\right)^{1/4} e^{-ikx} e^...

Sometimes in exercises we are asked to calculate the fourier series of a function. But there are two ways to do that. If $f:\Bbb R\mapsto\Bbb R$ is $T$-pe...

I factored $x^3 - 8 = 0$ and I only got $x = 2$, but the answer said it's $x=2$ and $x=-1 \pm \sqrt{3}i$? How do you get $x=-1 \pm\sqrt{3}i$?

I can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on '...