# Art museum of math formulas for pi

• 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 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 \tilde{f}:H\to K as follows. For each h\in H, […] • Determinant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […] • Rotation Matrix in Space and its Determinant and Eigenvalues For a real number 0\leq \theta \leq \pi, we define the real 3\times 3 matrix A by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.$ (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]