Rongzhou at 20:08, 3 October 2026

2026-10-03T20:08:44Z

← Older revision		Revision as of 20:08, 3 October 2026
Line 1:		Line 1:
	A '''linear system''' is a system of multiple [[linear equation\|linear equations]]		A '''linear system''' is a system of multiple [[linear equation\|linear equations]]

			== Matrix notation ==

			All linear systems can be rewritten in matrix form:

			{{#content:Q1977}}

			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 [[Linear equation\|homogenous]].

Rongzhou at 12:55, 2 October 2026

2026-10-02T12:55:53Z

← Older revision		Revision as of 12:55, 2 October 2026
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	~~== Overview ==~~		A '''linear system''' is a system of multiple [[linear equation\|linear equations]]

	a system of linear equations

2026-10-02T12:53:26Z

Created the wiki page for linear system

New page

== Overview ==

a system of linear equations