Linear system

A linear system is a system of multiple linear equations

Matrix notation

All linear systems can be rewritten in matrix form:

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

where:

• $A$ is the coefficient matrix
• $\mathbf{x}$ is the vector of unknowns
• $\mathbf{b}$ is the constant vector

For example, the system

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

can be represented as:

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

Equivalently, it can be represented in the augmented matrix form:

$$ \left[ \begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array} \right]. $$

Homogeneity

A linear system is homogenous if and only if each member equation is homogenous.