행렬과 연립방정식의 수식 표현

수학노트
이동: 둘러보기, 검색

\(\mathbf{A} = \begin{bmatrix} 9 & 8 & 6 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 0 & 5 \end{bmatrix}\)

 

  1. \mathbf{A} = \begin{bmatrix} 9 & 8 & 6 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 0 & 5 \end{bmatrix}

 

 

\(\mathbf{A} = \begin{pmatrix} 9 & 8 & 6 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 0 & 5 \end{pmatrix}\)

  1. \mathbf{A} = \begin{pmatrix} 9 & 8 & 6 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 0 & 5 \end{pmatrix}

 

 

\(\mathbf{A} = \begin{pmatrix} 9 & 8 \\ 1 & 2 \end{pmatrix}\)

  1. \mathbf{A} = \begin{pmatrix} 9 & 8 \\ 1 & 2  \end{pmatrix}

 

 

 

\(\Large A\ =\ \large\left( \begin{array}{c.cccc}&1&2&\cdots&n\\ \hdash1&a_{11}&a_{12}&\cdots&a_{1n}\\ 2&a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ n&a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right)\)

  1. \Large A\ =\ \large\left(\begin{array}{c.cccc}&1&2&\cdots&n\\         \hdash1&a_{11}&a_{12}&\cdots&a_{1n}\\         2&a_{21}&a_{22}&\cdots&a_{2n}\\         \vdots&\vdots&\vdots&\ddots&\vdots\\         n&a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right)

 

 

\(\begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& b_m. \\ \end{alignat}\)

  1. \begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& b_m. \\ \end{alignat}

 

 

\(A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\quad \bold{x}= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},\quad \bold{b}= \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}\)

  1. A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\quad \bold{x}= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},\quad \bold{b}= \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}

 

 

 

\(\mathbf{X}=\left(\begin{array}{ccc}x_{11} & x_{12} & \ldots \\x_{21} & x_{22} & \ldots \\\vdots & \vdots & \ddots\end{array} \right)\)

  1. \mathbf{X}=\left(\begin{array}{ccc}x_{11} & x_{12} & \ldots \\x_{21} & x_{22} & \ldots \\\vdots & \vdots & \ddots\end{array} \right)

 

 

 

\(\begin{array}{c.cccc}&1&2&\cdots&n\\ \hdash1&a_{11}&a_{12}&\cdots&a_{1n}\\ 2&a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ n&a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\)

  1. \begin{array}{c.cccc}&1&2&\cdots&n\\ \hdash1&a_{11}&a_{12}&\cdots&a_{1n}\\ 2&a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ n&a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}

 

 

\(\normalsize \left(\large\begin{array}{GC+23} \varepsilon_x\\\varepsilon_y\\\varepsilon_z\\\gamma_{xy}\\ \gamma_{xz}\\\gamma_{yz}\end{array}\right)\ {\Large=} \ \left[\begin{array}{CC} \begin{array}\frac1{E_{\fs{+1}x}} &-\frac{\nu_{xy}}{E_{\fs{+1}x}} &-\frac{\nu_{\fs{+1}xz}}{E_{\fs{+1}x}}\\ -\frac{\nu_{yx}}{E_y}&\frac1{E_{y}}&-\frac{\nu_{yz}}{E_y}\\ -\frac{\nu_{\fs{+1}zx}}{E_{\fs{+1}z}}& -\frac{\nu_{zy}}{E_{\fs{+1}z}} &\frac1{E_{\fs{+1}z}}\end{array} & {\LARGE 0} \\ {\LARGE 0} & \begin{array}\frac1{G_{xy}}&&\\ &\frac1{G_{\fs{+1}xz}}&\\&&\frac1{G_{yz}}\end{array} \end{array}\right] \ \left(\large\begin{array} \sigma_x\\\sigma_y\\\sigma_z\\\tau_{xy}\\\tau_{xz}\\\tau_{yz} \end{array}\right)\)

  1. \normalsize         \left(\large\begin{array}{GC+23}         \varepsilon_x\\\varepsilon_y\\\varepsilon_z\\\gamma_{xy}\\         \gamma_{xz}\\\gamma_{yz}\end{array}\right)\ {\Large=}         \ \left[\begin{array}{CC}         \begin{array}\frac1{E_{\fs{+1}x}}         &-\frac{\nu_{xy}}{E_{\fs{+1}x}}         &-\frac{\nu_{\fs{+1}xz}}{E_{\fs{+1}x}}\\         -\frac{\nu_{yx}}{E_y}&\frac1{E_{y}}&-\frac{\nu_{yz}}{E_y}\\         -\frac{\nu_{\fs{+1}zx}}{E_{\fs{+1}z}}&         -\frac{\nu_{zy}}{E_{\fs{+1}z}}         &\frac1{E_{\fs{+1}z}}\end{array} & {\LARGE 0} \\         {\LARGE 0} & \begin{array}\frac1{G_{xy}}&&\\         &\frac1{G_{\fs{+1}xz}}&\\&&\frac1{G_{yz}}\end{array}         \end{array}\right]         \ \left(\large\begin{array}         \sigma_x\\\sigma_y\\\sigma_z\\\tau_{xy}\\\tau_{xz}\\\tau_{yz}         \end{array}\right)

 

\(a^2 + b^2 &=& c^2\\ \frac{ab}{2} &=& n\)


  1. a^2 + b^2 &=& c^2\\ \frac{ab}{2} &=& n

 

 

 

\(\left((x)\right)= \begin{cases} x-\lfloor x\rfloor - 1/2 & \mbox{ if }x\in\mathbb{R}\setminus\mathbb{Z} \\ 0 & \mbox{ if } x\in\mathbb{Z} \end{cases}\)

  1. \left((x)\right)= \begin{cases} x-\lfloor x\rfloor - 1/2 & \mbox{ if }x\in\mathbb{R}\setminus\mathbb{Z} \\ 0 & \mbox{ if } x\in\mathbb{Z} \end{cases}

 

 

\(g_1(\chi) = \begin{cases} \sqrt{p}, & p \equiv 1 \pmod{4}, \\ i \sqrt{p}, & p \equiv 3 \pmod{4}. \end{cases}\)

  1. g_1(\chi) = \begin{cases} \sqrt{p}, & p \equiv 1 \pmod{4}, \ i \sqrt{p}, & p \equiv 3 \pmod{4}. \end{cases}

 

 

\(\left(\frac{a}{p}\right) = \begin{cases} \;\;\,0\mbox{ if } a \equiv 0 \pmod{p} \\+1\mbox{ if }a \not\equiv 0\pmod{p} \mbox{ and for some integer }x, \;a\equiv x^2\pmod{p} \\-1\mbox{ if there is no such } x. \end{cases}\)

  1. \left(\frac{a}{p}\right) = \begin{cases} \;\;\,0\mbox{ if } a \equiv 0 \pmod{p} \\+1\mbox{ if }a \not\equiv 0\pmod{p} \mbox{ and for some integer }x, \;a\equiv x^2\pmod{p} \\-1\mbox{ if there is no such } x. \end{cases}

 

 

\(\tilde y=\left\{ {\ddot x\text{ if UNIQba85fc59c067b1ac-MathJax-1-QINU odd}\atop\hat{\,\bar x+1}\text{ if even}}\right\)

  1. \tilde y=\left\{ {\ddot x\text{ if $\vec x$ odd}\atop\hat{\,\bar x+1}\text{ if even}}\right

 

 

 

\(\begin{eqnarray}\int_{0}^{\frac{\pi}{2}}\frac{\theta^{2}}{\sin\theta}\, d\theta & = &\sum_{n=1}^{\infty}\frac{4^{n-1}}{n^{2}\binom{2n}{n}}\int_{0}^{\frac{\pi}{2}}2\sin^{2n-1}\theta\, d\theta \nonumber \\ & = &\sum_{n=1}^{\infty}\frac{4^{2n-1}}{n^{3}\binom{2n}{n}^{2}} \nonumber \\ & = &\sum_{n=1}^{\infty}\frac{1}{n} \left[ \int_{0}^{\frac{\pi}{2}} \sin^{2n-1} t \, dt \right] \left[ \int_{0}^{\frac{\pi}{2}} \sin^{2n-1} u \, du \right] \nonumber \\ & = & - \int_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} \frac{\log(1 - \sin^2 t \sin^2 u)}{\sin t \sin u} \, dt du \nonumber \\ & = &-\int_{0}^{1}\int_{0}^{1}\frac{\log(1-x^{2}y^{2})}{xy\sqrt{1-x^{2}}\sqrt{1-y^{2}}}\, dxdy \nonumber \end{eqnarray} \)

  1. \begin{eqnarray}\int_{0}^{\frac{\pi}{2}}\frac{\theta^{2}}{\sin\theta}\, d\theta & = &\sum_{n=1}^{\infty}\frac{4^{n-1}}{n^{2}\binom{2n}{n}}\int_{0}^{\frac{\pi}{2}}2\sin^{2n-1}\theta\, d\theta \nonumber \\ & = &\sum_{n=1}^{\infty}\frac{4^{2n-1}}{n^{3}\binom{2n}{n}^{2}} \nonumber \\ & = &\sum_{n=1}^{\infty}\frac{1}{n} \left[ \int_{0}^{\frac{\pi}{2}} \sin^{2n-1} t \, dt \right] \left[ \int_{0}^{\frac{\pi}{2}} \sin^{2n-1} u \, du \right] \nonumber \\ & = & - \int_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} \frac{\log(1 - \sin^2 t \sin^2 u)}{\sin t \sin u} \, dt du \nonumber \ & = &-\int_{0}^{1}\int_{0}^{1}\frac{\log(1-x^{2}y^{2})}{xy\sqrt{1-x^{2}}\sqrt{1-y^{2}}}\, dxdy \nonumber \end{eqnarray} 

 

 

\(\begin{align} s(x) &= \sum_{k=0}^{\infty} F_k x^k \\ &= F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \\ &= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \\ &= x + x\sum_{k=0}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \\ &= x + x s(x) + x^2 s(x) \end{align}\)

  1. \begin{align} s(x) &= \sum_{k=0}^{\infty} F_k x^k \ &= F_0 + F_1x + \sum_{k=2}^{\infty} \left( F_{k-1} + F_{k-2} \right) x^k \ &= x + \sum_{k=2}^{\infty} F_{k-1} x^k + \sum_{k=2}^{\infty} F_{k-2} x^k \ &= x + x\sum_{k=0}^{\infty} F_k x^k + x^2\sum_{k=0}^{\infty} F_k x^k \ &= x + x s(x) + x^2 s(x) \end{align}

 

\((1) \quad \int_{0}^{\infty} \frac{1}{x^2} dx\)