Linear Algebra, 5th Edition

3.3 Systems of Linear Equations—Theoretical Aspects

This section and the next are devoted to the study of systems of linear equations, which arise naturally in both the physical and social sciences. In this section, we apply results from Chapter 2 to describe the solution sets of systems of linear equations as subsets of a vector space. In Section 3.4, we will use elementary row operations to provide a computational method for finding all solutions to such systems.

The system of equations

$$\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=b_2\\ (S)⋮\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n=b_m,\end{array}$$

where $a_{ij}$ and $b_i$ ($1\le i\le m$ and $1\le j\le n$) are scalars in a field F and $x_1,x_2,\dots ,x_n$ are n variables taking values in F, is called a system of m linear equations in n unknowns over the field ...