12.6 Coefficient Matrix of a Linear System

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Coefficient Matrix of a Linear System

Given a linear system of $m$ equations and $n$ unknowns

\begin{aligned} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} & = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} & = b_{2} \\ ⋮ \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} & = b_{m} \end{aligned}

we can write this system as a linear systems problem $A \vec{x} = \vec{b}$ where the matrix

A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}]

is the coefficient matrix of our linear system.

Row dimension of matrix $A$ indicates number of equations we are solving at the same time
Column dimension of matrix $A$ counts the number of unknown variables

Remark. The basic idea of solving a linear system of equations is to transform the linear system into an equivalent matrix equation and transform the matrix so the linear system is much easier to solve.
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