“우리 대부분은 초라한 옷차림과 엉터리 가구들을 부끄럽게 여기지만, 그보다는 초라한 생각과 엉터리 철학을 부끄럽게 여길 줄 알아야 한다.” – Albert Einstein

19800101
math

euclidean algorithm

최대공약수 찾는 방법중 하나

integer factorization

최대공약수, 최소공배수 찾는 방법중 하나

sieve of eratosthenes

소수 찾는 방법

mathematical induction

자연수에 관한 명제 \(P(n)\)이 모든 자연수(또는, 어떤 자연수보다 큰 모든 자연수)에 대하여 성립함을 보이는 증명법


\(\pi\)

\(3.14\;159\;265\;358\;979\;323\;238\;462\;643\;383\dotsc\)

\(e\)

\(2.71\;828\;182\;845\;904\;523\;536\;028\;747\;135\dotsc\)

fibonacci number

\(F_0=0,\;F_1=1,\;F_{n+2}=F_{n+1}+F_n\)

\(0\;1\;1\;2\;3\;5\;8\;13\;21\;34\;55\;89\;144\dotsc\)

factorial

\(0! = 1\)

\(1! = 1 * 0! = 1\)

\(2! = 2 * 1! = 2\)

\(3! = 3 * 2! = 6\)

\(4! = 4 * 3! = 24\)

\(5! = 5 * 4! = 5 * 4 * 3 * 2 * 1 = 120\)

exponentiation

\(a^n = (1 * a \ldots a * a)\; n\)번만큼 \(a\)를 곱

\(a^0 = 1\)

\(a^1 = a\)

\(a^b * a^c = a^{b+c}\)

\(a^m \div a^n = a^{m-n}\)

\((a^n)^m = a^{nm}\)

\((a^b)^c = (a^c)^b\)

\(a^c * b^c = (ab)^c\)

\(a^{n+1} = a^n * a\)

\(a^{-n} = {\Large{1 \over a^n}}\)

\(a^n * a^m = a^{n+m} ∴ a^n * a^{-m} = a^{n+(-m)}\)

\(2^1 * 2^{-1} = 2 * {\Large{1 \over 2}} = 1 ∴ a^{1+(-1)} = a^0 = 1 \; (a \ne 0)\)

\(({\Large{a \over b}})^{-2} = ({\Large{1 \over{a \over b}^2}}) = ({\Large{b \over a}})^2\)

\(8^3 = 2^3 * 4^3\)

\(2^n\)

0 1 / 1 2 / 2 4 / 3 8 / 4 16 / 5 32 / 6 64 / 7 128 / 8 256 / 9 512 / 10 1,024 / 11 2,048 / 12 4,096

13 8,192 / 14 16,384 / 15 32,768 / 16 65,536 / 17 131,072 / 18 262,144 / 19 542,288 / 20 1,048,576

21 2,097,152 / 22 4,194,304 / 23 8,388,608 / 24 16,777,216 / 25 33,554,432 / 26 67,108,864

27 134,217,728 / 28 268,435,456 / 29 536,870,912 / 30 1,073,741,824 / 31 2,147,483,648 / 32 4,294,967,296

\(8^n\)

0 1 / 1 8 / 2 64 / 3 512 / 4 4,096 / 5 32,768 / 6 262,144 / 7 2,097,152 / 8 16,777,216 / 9 134,217,728 / 10 1,073,741,824

radian degree

\(1\;rad = {\Large{180^\circ \over \pi}} \approx 57.2958^\circ\)

\(1^\circ = {\Large{\pi \over 180^\circ}} \approx 0.0175\;rad\)

logarithm

if ( a > 0 && a != 1 && y > 0 ) \((y = a^x) == (log_a y = x)\)


square root

\(\sqrt{u\cdot u*v\cdot v}=\sqrt{u\cdot u}*\sqrt{v\cdot v}\)

magnitude(norm)

\(||u|| = \sqrt{u\cdot u} = \sqrt{u_x*u_x+u_y*u_y}\)

normalization

\(\hat{u} = {\Large {u \over ||u||}}\)

dot product

\(u \cdot v = u_x v_x + u_y v_y + u_z v_z\)

\(\theta= \arccos(\hat{u}\cdot\hat{v})\)

\(w\cdot(u\times v)>0.0\text{ ? cw : ccw // } w\) is an world up vector, left hand coordinate

\(cos\theta=\hat{u}\cdot\hat{v}={\Large{u\over \sqrt{u\cdot u}}}\cdot{\Large{v\over \sqrt{v\cdot v}}}={\Large{u\over \sqrt{u\cdot u}}}*{\Large{u\over \sqrt{u\cdot u}}}+{\Large{v\over \sqrt{v\cdot v}}}*{\Large{v\over \sqrt{v\cdot v}}}={\Large{u\cdot v\over \sqrt{(u\cdot u)\cdot(v\cdot v)}}}\)

cross product

\(u\times v=(u_y v_z-u_z v_y,u_z v_x-u_x v_z,u_x v_y-u_y v_x)\)

\(u\times(v+w)=(u\times v)+(u\times w)\)

\(v\times 0=0\times v=0\)

\(v\times v=0\)

triple product

\(u\cdot(v\times w)=v\cdot(u\times w)=w\cdot(u\times v)=det(u,\;v,\;w)\)

projection

\(u\cdot v=||u||||v||\cos\theta={\Large{||u||||v||||u_{proj}||\over ||u||}}=||v||||u_{proj}||\)

\(||u_{proj}||={\Large{u\cdot v\over ||v||}}\)

\(u_{proj}=||u_{proj}||*\hat{v}={\Large{u\cdot v\over ||v||}}*{\Large{v\over ||v||}}={\Large{u\cdot v\over ||v||^2}}*v\)

\(u_{proj}=(u\cdot \hat{v})*\hat{v}\)


matrix multiplication

\(\begin{bmatrix} A_1 & A_2 \\ A_3 & A_4 \end{bmatrix} * \begin{bmatrix} B_1 & B_2 \\ B_3 & B_4 \end{bmatrix} = \begin{bmatrix} (A_1,A_2) \cdot (B_1,B_3) & (A_1,A_2) \cdot (B_2,B_4) \\ (A_3,A_4) \cdot (B_1,B_3) & (A_3,A_4) \cdot (B_2,B_4) \end{bmatrix}\)

matrix minor

\(A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \quad \bar{A}_{11} = [ A_{22} ]\)

determinant

\(\text{det}A \ne 0 \to\) this is an invertible matrix

\(\text{det}A = \displaystyle\sum_{j=1}^{n} A_{1j} (-1)^{1+j} \text{det}\bar{A}_{1j}\)

\(\text{det}[A_{11}] = A_{11}\)

\(\text{det}\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\)

\(= A_{11}\text{det}[A_{22}] – A_{12}\text{det}[A_{21}]\)

\(= A_{11}A_{22} – A_{12}A_{21}\)

\(\text{det}\begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}\)

\(= A_{11}\begin{bmatrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{bmatrix} – A_{12}\begin{bmatrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{bmatrix} + A_{13}\begin{bmatrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{bmatrix}\)

\(= A_{11}A_{22}A_{33} + A_{12}A_{23}A_{31} + A_{13}A_{21}A_{32} – A_{13}A_{22}A_{31} – A_{12}A_{21}A_{33} – A_{11}A_{23}A_{32}\)

\(A = \begin{bmatrix} 2 & -5 & 3 \\ 1 & 3 & 4 \\ -2 & 3 & 7 \end{bmatrix}\)

\(\text{det}A = (2 * 3 * 7) + (-5 * 4 * -2) + (3 * 1 * 3) – (3 * 3 * -2) – (-5 * 1 * 7) – (2 * 4 * 3) = 42+40+9+18+35-24=120\)

finding determinant by gauss jordan elimination

initial matrix

\(\begin{bmatrix} (2) & 1 & 3 \\ 4 & 3 & 1 \\ 2 & 4 & 1 \end{bmatrix}\)

multiply the 1st row by 2

\(\begin{bmatrix} 4 & 2 & 6 \\ 4 & 3 & 1 \\ 2 & 4 & 1 \end{bmatrix}\)

subtract the 1st row from the 2nd row and restore 1st row

\(\begin{bmatrix} (2) & 1 & 3 \\ 0 & (1) & -5 \\ 2 & 4 & 1 \end{bmatrix}\)

subtract the 1st row from the 3rd row

\(\begin{bmatrix} (2) & 1 & 3 \\ 0 & (1) & -5 \\ 0 & 3 & -2 \end{bmatrix}\)

multiply the 2nd row by 3

\(\begin{bmatrix} (2) & 1 & 3 \\ 0 & 3 & -15 \\ 0 & 3 & -2 \end{bmatrix}\)

subtract the 2nd row from the 3rd row and restore 2nd row

\(\begin{bmatrix} (2) & 1 & 3 \\ 0 & (1) & -5 \\ 0 & 0 & (13) \end{bmatrix}\)

multiply the main diagonal elements

\(2 * 1 * 13 = 26\)

cofactor

\(C_{ij} = (-1)^{i+j}\text{det}\bar{A}_{ij}\)

cofactor matrix

\(C_A = \begin{bmatrix} C_{11} & C_{12} & \ldots & C_{1j} \\ C_{21} & C_{22} & \ldots & C_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ C_{i1} & C_{i2} & \ldots & C_{ij} \end{bmatrix}\)

adjugate matrix

\(\text{adj}A = {C_A}^T\)

inverse matrix

\(A^{-1} = {\Large{\text{adj}A \over \text{det}A}}\)

scaling matrix

\(S = \begin{bmatrix} S_x & 0 & 0 \\ 0 & S_y & 0 \\ 0 & 0 & S_z \end{bmatrix} S^{-1} = \begin{bmatrix} {1 \over S_x} & 0 & 0 \\ 0 & {1 \over S_y} & 0 \\ 0 & 0 & {1 \over S_z} \end{bmatrix}\)

rotation matrix

\(C = \cos\theta, S = \sin\theta\)

\(S = \begin{bmatrix} C + ( 1 – C ) x^2 & ( 1 – c ) xy + Sz & ( 1 – c ) xz – Sy \\ ( 1 – c ) xy – Sz & C + ( 1 – C ) y^2 & ( 1 – c ) yz + Sx \\ ( 1 – c ) xz + Sy & ( 1 – c ) yz – Sx & C + ( 1 – C ) z^2 \end{bmatrix}\)

\(R_x = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & C & S & 0 \\ 0 & -S & C & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} R_y = \begin{bmatrix} C & 0 & -S & 0 \\ 0 & 1 & 0 & 0 \\ S & 0 & C & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} R_z = \begin{bmatrix} C & S & 0 & 0 \\ -S & C & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\)

translation matrix

\(T = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ x & y & z & 1 \end{bmatrix} T^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -x & -y & -z & 1 \end{bmatrix}\)

change of basis

\(U, V, W = \text{new basis}, Q = \text{new origin}\)

\(B = \begin{bmatrix} U_x & U_y & U_z & 0 \\ V_x & V_y & V_z & 0 \\ W_x & W_y & W_z & 0 \\ Q_x & Q_y & Q_z & 1 \end{bmatrix}\)

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