Art museum of math formulas for pi

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• Beautiful Formulas for pi=3.14… The number $\pi$ is defined a s the ratio of a circle's circumference $C$ to its diameter $d$: $\pi=\frac{C}{d}.$ $\pi$ in decimal starts with 3.14... and never end. I will show you several beautiful formulas for $\pi$.   Art Museum of formulas for $\pi$ […]
• Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$ Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying $T\left(\, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\,\right)=\begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} […] • The Rotation Matrix is an Orthogonal Transformation Let \mathbb{R}^2 be the vector space of size-2 column vectors. This vector space has an inner product defined by  \langle \mathbf{v} , \mathbf{w} \rangle = \mathbf{v}^\trans \mathbf{w}. A linear transformation T : \R^2 \rightarrow \R^2 is called an orthogonal transformation if […] • Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018. Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is \[2018=2\cdot 1009.$ Here $2$ and $1009$ are […]
• The Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
• The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements Let $G$ be an abelian group. Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively. Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$. Also determine whether the statement is true if $G$ is a […]
• The Cyclotomic Field of 8-th Roots of Unity is $\Q(\zeta_8)=\Q(i, \sqrt{2})$ Let $\zeta_8$ be a primitive $8$-th root of unity. Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.   Proof. Recall that the extension degree of the cyclotomic field of $n$-th roots of unity is given by […]
• A Group Homomorphism that Factors though Another Group Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$. Define a map $\bar{f}:H\to K$ as follows. For each […]