System of Linear Equations

Chapter 1 System of Linear Equations

In this chapter, we shall deal with solutions of system of linear equations. We shall also introduce, row operations and how it helps in solving a system of linear equation. Row operations will be used as one of the tools for solving a system. Let us first define what do we mean by a sytem of linear equations.

Definition 1.0.1.

A system of linear equations is a finite collection of linear equations. Consider a system of \(m\) linear equations in \(n\)-variables \(x_1,\ldots, x_n\text{.}\)

\begin{align} a_{11}x_1 + a_{12}x_2+ \ldots + a_{1n} x_n \amp= \amp b_1\notag\\ a_{21}x_1 + a_{22}x_2+ \ldots + a_{2n} x_n \amp= \amp b_2\notag\\ \vdots \amp \amp \vdots \tag{1.0.1}\\ a_{m-11}x_1 + a_{m-12}x_2+ \ldots + a_{m-1n} x_n \amp = \amp b_{m-1}\notag\\ a_{m1}x_1 + a_{m2}x_2+ \ldots + a_{mn} x_n \amp = \amp b_m\notag \end{align}

Here \(a_{ij}\in \R\) for \(1\leq i\leq m,1\leq j\leq n\) and \(b_i\in \R\) for \(1\leq i\leq m\text{.}\) The above system of linear equations can be represented in compact form using the summation notation as follows:

\begin{equation} \sum_{j=1}^n a_{ij}x_j =b_i,\quad 1\leq i\leq m\text{.}\tag{1.0.2} \end{equation}

Next let us define, what is meaning of a solution and the solution set of the system of linear system of equations (1.0.1).

Definition 1.0.2.

A solution of the system of linear equation is an \(n\)-tuple of the real numbers \((y_1, y_2, \cdots, y_n)\) such that \(x_i = y_i\) for each \(i = 1, 2, \ldots, n\) satisfies each of the \(m\)-equations. The set of all solutions of a system is called the solution set. It is a subset of \(\R^n\text{,}\) the Cartesian product of the set \(\R\) of real numbers \(n\)-times. If we denote it by \(S\subset \R^n\text{,}\) then

\begin{equation*} S=\{(x_1,\ldots,x_n)\in \R^n:\sum_{j=1}^n a_{ij}x_j =b_i,\quad 1\leq i\leq m\}\text{.} \end{equation*}

If the solution set is nonempty then the system is said to be consistent and otherwise it is said to be inconsistent.

The above system of linear equation can be written as a single matrix equation as \(AX = B\text{,}\) where

\begin{equation} A=\begin{pmatrix} a_{11} \amp a_{12} \amp \ldots \amp a_{1n} \\ a_{21} \amp a_{22} \amp \ldots \amp a_{2n} \\ \vdots\amp \vdots\amp \vdots\amp \vdots\\ a_{m1} \amp a_{m2} \amp \ldots \amp a_{mn} \end{pmatrix}, X=\begin{pmatrix}x_1\\x_2\\ \vdots\\ x_n \end{pmatrix}, B=\begin{pmatrix}b_1\\b_2\\ \vdots\\ b_m \end{pmatrix} \tag{1.0.3} \end{equation}

The matrix \(A\) is called the coefficient matrix, \(B\) is called the column matrix of constants or known terms and \(X\) is called the column matrix of the unknowns. All the essential data of the system can be gathered in one matrix by inserting the column matrix of the constants to the coefficient matrix as \(n+1\)-th column. This matrix is called the augmented or enlarged matrix of the system. Once we know the system, we can get the augmented matrix associated with the system and conversely once we know the augmented matrix, we know the system and start calculating the solutions.

The augmented matrix of the above system \(AX=B\) is given by

\begin{equation} [A~|~B]= \begin{pmatrix} a_{11} \amp a_{12} \amp \cdots \amp a_{1n} \amp | \amp b_1\\ a_{21} \amp a_{22} \amp \cdots \amp a_{2n} \amp | \amp b_2\\ \vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots \amp \vdots\\ a_{m1} \amp a_{m2} \amp \cdots \amp a_{mn} \amp | \amp b_m\\ \end{pmatrix}\tag{1.0.4} \end{equation}

Definition 1.0.3.

Two systems of linear equations are called equivalent if and only if they have the same set of solutions.

We will define some basic operations by which a system is transformed into an equivalent system.

Geometrically, solving a system of linear equations in two (or three) unknowns is equivalent to determining whether or not a family of lines (or planes) has a common point of intersection.

Remark 1.0.4.

Let \(A\) be \(n\times n\) real matrix whose columns are \(a_1,\ldots, a_n\) and \(x=\begin{pmatrix}x_1\amp\cdots \amp x_n\end{pmatrix}^T\text{.}\) Then \(Ax=x_1a_1+x_2a_2+\cdots +a_nx_n\text{.}\) In particular, \(Ax\) is a linear combinations of column of \(A\text{.}\) Thus if \(\alpha=\begin{pmatrix}\alpha_1\amp\cdots \amp \alpha_n\end{pmatrix}^T\) is a solution of \(Ax=b\text{,}\) then \(b\) is linear combination of columns of \(A\text{.}\) For example,

\begin{equation*} AX= \begin{pmatrix} 2 \amp 3 \amp 1 \amp 4\\ 1 \amp 2 \amp -1 \amp 3\\ 3 \amp -2 \amp 0 \amp 2\\ -2 \amp 0 \amp 2 \amp 1 \end{pmatrix} \begin{pmatrix} x_1\\x_2\\x_3\\x_4\end{pmatrix} =\begin{pmatrix} 2 x_1+ 3x_2 + x_3 + 4x_4\\ x_1 +2x_2 -x_3+ 3x_3\\ 3 x_1 -2 x_2+2x_4\\ -2x_1 + 2x_3 +x_4 \end{pmatrix}. \end{equation*}

This can be written as

\begin{equation*} AX= x_1 \begin{pmatrix} 2\\1\\3\\-2\end{pmatrix}+ x_2 \begin{pmatrix} 3\\2\\-2\\0\end{pmatrix}+ x_3 \begin{pmatrix} 1\\-1\\0\\2\end{pmatrix}+ x_4 \begin{pmatrix} 4\\3\\2\\1\end{pmatrix}. \end{equation*}

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