Linear Algebra with Applications, 10th Edition

1.1 Systems of Linear Equations

A linear equation in n unknowns is an equation of the form

$$a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n=b$$

where $a_1,a_2,\dots ,a_n$ and b are real numbers and $x_1,x_2,\dots ,x_n$ are variables. A linear system of m equations in n unknowns is then a system of the form

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

where the $a_{\mathit{ij}}\text{’s}$ and the $b_i\text{’s}$ are all real numbers. We will refer to systems of the form (1) as $m\times n$ linear systems. The following are examples of linear systems:

1. $\begin{array}{r}{x}_1+2x_2=5\\ 2x_1+3x_2=8\end{array}$

2. $\begin{array}{r}{x}_1-x_2+x_3=2\\ 2x_1+x_2-x_3=4\end{array}$

3. $\begin{array}{l}{x}_1+x_2=2\\ x_1-x_2=1\\ x_1\phantom{-}\phantom{x_2}=4\end{array}$

System (a) is a $2\times 2$ system, (b) is a $2\times 3$ system, and (c) is a $3\times 2$ system.

By a solution of an $m\times n$ system, we mean an ordered n-tuple of numbers $(x_1,x_2,\dots ,x_n)$ that satisfies all the equations of ...