- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.17 Systems of linear equations
$\quad$ Let $A = (a_{ij})$ be a given $m \times n$ matrix of numbers, and let $c_1, \dots, c_m$ be $m$ further numbers. A set of $m$ equations of the form \begin{align*} (2.23) \qquad \sum_{k=1}^n a_{ik}x_k = c_i \qquad \text{for} \quad i = 1, 2, \dots, m \end{align*} is called a system of $m$ linear equations in $n$ unknowns. Here, $x_1, \dots, x_n$ are the unknowns. A solution of the system is any $n$-tuple of numbers $(x_1, \dots, x_n)$ for which all the equations are satisfied. The matrix $A$ is called the coefficient matrix of the system.
$\quad$ Choosing the usual bases of unit coordinate vectors in $V_n$ and $V_m,$ the coefficient matrix $A$ determines a linear transformation $T: V_n \to V_m,$ which maps an arbitrary vector $x = (x_1, \dots, x_n)$ in $V_n$ onto $y = (y_1, \dots, y_m)$ in $V_m.$ The components of $y$ are given by the $m$ linear equations: \begin{align*} y_i &= \sum_{k=1}^m a_{ik}x_k \qquad \text{for} \quad i = 1, \dots, m \end{align*} If we let $c = (c_1, \dots, c_m)$ be the vector in $V_m$ whose components appear in (2.23), we can write the system of linear equations as \begin{align*} T(x) &= c. \end{align*} The system has a solution if and only if $c$ is in the range of $T.$
$\quad$ With each linear system (2.23), we can associate another system \begin{align*} \sum_{k=1}^n a_{ik}x_k &= 0 \qquad \text{for} \quad i = 1, \dots, m \end{align*} obtained by replacing each $c_i$ in (2.23) by $0.$ This is called the homogeneous system corresponding to (2.23). If $c \neq O,$ (2.23) is called a nonhomogeneous system. A vector $x$ in $V_n$ will satisfy the homogeneous system if and only if $$T(x) = O,$$ where $T$ is the linear transformation determined by the coefficient matrix. The set of solutions to the homogeneous system is the null space of $T.$ The next theorem will describe the relation between the solutions of the homogeneous system and the nonhomogeneous system.
$\quad$ Theorem 2.18. $\quad$ Assume the nonhomogeneous system (2.23) has a solution, say $b.$
(a) $\quad$ If a vector $x$ is a solution of the nonhomogeneous system, then the vector $v = x - b$ is a solution of the corresponding homogeneous system.
(b) $\quad$ If a vector $v$ is a solution of the homogeneous system, then the vector $x = v + b$ is a solution of the nonhomogeneous system.
$\quad$ Proof. $\quad$ Let $T: V_n \to V_m$ be the linear transformation described by (2.23) \begin{align*} (2.23) \qquad \sum_{k=1}^n a_{ik}x_k = c_i \qquad \text{for} \quad i = 1, 2, \dots, m \end{align*} If $b$ is a solution to the nonhomogeneous system, $T(b) = c,$ where $c \neq O$ is the element in $V_m$ whose components are given by the system of linear equations in (2.23). Then, if $v$ and $x$ are vectors in $V_n$ such that $v = x - b,$ we have \begin{align*} T(v) &= T(x - b) = T(x) - T(b) = T(x) - c \end{align*} Then, $T(v) = O$ if and only if $T(x) = c.$ That is, $v$ is a solution to the homogeneous system if and only if $x$ is a solution to the nonhomogeneous system. This proves both (a) and (b).
$\quad$ This theorem shows that the problem of finding the solutions of a nonhomogeneous system splits naturally into two parts:
(1) $\quad$ Finding all solutions $v$ of the homogeneous system $($that is, determining the null space of $T)$
(2) $\quad$ Finding one particular solution $b$ of the nonhomogeneous system.
$\quad$ Let $k = \dim N(T).$ If we can find $k$ independent solutions $v_1, \dots, v_k$ of the homogeneous system, they will form a basis for $N(T),$ and we can form every $v$ in $N(T)$ by forming all possible linear combinations \begin{align*} v &= t_1v_1 + \cdots + t_kv_k, \end{align*} where $t_1, \dots, t_k$ are arbitrary scalars. This linear combination is called the general solution of the homogeneous system. If $b$ is one particular solution to the nonhomogeneous system, then all solutions $x$ are given by \begin{align*} x &= b + t_1v_1 + \cdots + t_kv_k \end{align*} This linear combination is called the general solution of the nonhomogeneous system. Now, we can restate Theorem 2.18 as follows:
$\quad$ Theorem 2.19. $\quad$ Let $T: V_n \to V_m$ be the linear transformation such that $T(x) = y,$ where $x = (x_1, \dots, x_n),$ $y = (y_1, \dots, y_m),$ and \begin{align*} y_i &= \sum_{k=1}^n a_{ik}x_k \qquad \text{for} \quad i=1, \dots, m. \end{align*} Let $k$ denote the nullity of $T.$ If $v = v_1, \dots, v_k$ are $k$ independent solutions of the nonhomogeneous systems $T(x) = O,$ and if $b$ is one particular solution of the nonhomogeneous system $T(x) = c,$ then the general solution of the nonhomogeneous system is \begin{align*} x &= b + t_1v_1 + \cdots + t_kv_k \end{align*} where $t_1, \dots, t_k$ are arbitrary scalars.