Linear Equation

14 Jan 2018 on Math, Linear algebra

System of linear equations

1. general form

${\begin{aligned}a_{11}x_{1}&+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}&+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots &\\a_{m1}x_{1}&+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m},\end{aligned}}$

2. Vector Equation

$x_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\cdots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}$

3. Matrix equation

$A{\mathbf {x}}={\mathbf {b}}$

$A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad {\mathbf {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad {\mathbf {b}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}$

4. matrix solution

${\mathbf {x}}=A^{-1}{\mathbf {b}}$

5. Properties

- dependent

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- independent

alt text

Reference

System_of_linear_equations - wikipedia