Systems of Linear Equations

Overview

Let $c_{1}, \dots, c_{n} \in R$ be constants. A system of linear equations is a collection of linear equations of the following form:

\begin{aligned} A_{1, 1} x_{1} + \dots + A_{1, m} x_{m} & = c_{1} \\ A_{2, 1} x_{1} + \dots + A_{2, m} x_{m} & = c_{2} \\ ⋮ \\ A_{n, 1} x_{1} + \dots + A_{n, m} x_{m} & = c_{n} \end{aligned}

Such a system is said to be homogeneous if $c_{1} = \dots = c_{n} = 0$ . Otherwise it is inhomogeneous.

Characterization

Given standard basis $e_{1}, \dots, e_{m} \in R^{m}$ , basis characterization shows the above linear system corresponds to a linear map $T \in L (R^{m}, R^{n})$ such that

\begin{aligned} T e_{1} & = ⟨ A_{1, 1}, A_{2, 1}, \dots, A_{n, 1} ⟩ \\ T e_{2} & = ⟨ A_{1, 2}, A_{2, 2}, \dots, A_{n, 2} ⟩ \\ ⋮ \\ T e_{m} & = ⟨ A_{1, m}, A_{2, m}, \dots, A_{n, m} ⟩ \end{aligned}

A homogeneous system of linear equations with more variables than equations has nonzero solutions. An inhomogenous system of linear equations with more equations than variables has no solution for some choice of the constant terms.