# Section 1.1 Linear Equations

Definition: 1. A **linear equation** in the variables, [latex]x_{1},...,x_{n}[/latex] is an equation that can be written in the form [latex]a_{1}x_{1}+....+a_{n}x_{n}=b[/latex] where [latex]a_{1},...,a_{n},b[/latex] are constants, real numbers or complex numbers.

2. A **system of linear equations** (or a linear system) is a collection of one or more linear equations involving the same variables, [latex]x_{1},....,x_{n}[/latex].

3. A** solution of the system** is a list of numbers, [latex]s_{1},...,s_{n}[/latex], that makes each equation a true statement when the values [latex]s_{1},...,s_{n}[/latex] are substituted for [latex]x_{1},...,x_{n}[/latex] respectively.

4. **Solve a system** means “find all solutions to the system.” The set of all possible solutions is called the **solution set of the linear system.**

5. Two linear systems are called **equivalent** if they have the same solution set.

**Theorem(fact): A system of linear equations has no solution, or exactly one solution, or infinitely many solutions.**

Definition: A system of linear equations is said to be **consistent** if it has either one solution or infinitely many solutions. A system of linear equations is said to be **inconsistent** if it has no solution.

Definition: 1. The essential information of a linear system can be recorded compactly in a rectangular array called a **matrix**.

2. The matrix associated to a linear system is called the **coefficient matrix** of the linear system.

3. An **augmented matrix** of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.

**Exercise 1**: Write down the coefficient matrix and the augmented matrix of the linear system.

[latex]\begin{array}{ccc} 2x_{2}-3x_{3}+x_{4} & = & 1\\ 3x_{1}-x_{3}+4x_{4} & = & 2\\ x_{1}+x_{2} & = & 3\\ x_{2}-x_{3}+x_{4} & = & 1 \end{array}[/latex]

Definition: The size of a matrix tells how many rows and columns it has. If [latex]m[/latex] and [latex]n[/latex] are positive numbers, an [latex]m\times n[/latex] matrix is a rectangular array of numbers with [latex]m[/latex] rows and [latex]n[/latex] columns.(**The number of rows always comes first.**)

Definition: 1. **Elementary row operations** include the following:

A. (Replacement) Replace one row by the sum of itself and a multiple of another row.

B. (Interchange) Interchange two rows.

C. (Scaling) Multiply all entries in a row by a nonzero constant.

2. Two matrices are called **row equivalent** if there is a sequence of elementary row operations that transforms one matrix into the other.

Facts: 1. Row operations are reversible.

2. If the augmented matrices of two linear systems are row equivalent, then the two systems are equivalent, i.e. they have the same solution set.

Questions: Given a linear system 1. Is the system consistent? 2. If the system has a solution, is the solution unique?

Example 2: Determine if the following system is consistent.

[latex]\begin{array}{ccc} x_{2}-x_{3} & = & 4\\ 3x_{1}+x_{3} & = & 1\\ 2x_{1}-x_{2} & = & 2 \end{array}[/latex]

**Exercise 2**: Determine if the following system is consistent.

[latex]\begin{array}{ccc} 2x_{2}-3x_{3} & = & 1\\ 3x_{1}-x_{3} & = & 2\\ x_{1}+x_{2} & = & 3 \end{array}[/latex]

**Example 3**: Determine if the following system is consistent.

[latex]\begin{array}{ccc} x_{1}+x_{2}-3x_{3} & = & 4\\ x_{1}-2x_{3} & = & 1\\ -x_{2}+x_{3} & = & 2 \end{array}[/latex]

**Exercise 3:** Determine if the following system is consistent.

[latex]\begin{array}{ccc} -x_{1}+x_{2}-3x_{3} & = & 1\\ x_{2}+3x_{3} & = & 2\\ x_{1}-2x_{2} & = & 3 \end{array}[/latex]

Group Work 1: Do the following three lines have a common point of intersection?

[latex]\begin{array}{cc} 2x_{1}-x_{2} & =1\\ x_{1}+x_{2} & =2\\ -x_{1}+3x_{2} & =3 \end{array}[/latex]

Group Work 2: Do the following three planes have a common point of intersection?

[latex]\begin{array}{cc} 2x_{1}-x_{2}+x_{3} & =1\\ x_{1}+x_{2} & =2\\ x_{2}+3x_{3} & =3 \end{array}[/latex]

Group Work 3: Find a [latex]h[/latex] such that the matrix is the augmented matrix of a consistent linear system.

[latex]\left[\begin{array}{ccc} 2 & -3 & 4\\ -4 & h & 1 \end{array}\right][/latex]

Group Work 4: Find a [latex]h[/latex] such that the matrix is the augmented matrix of an inconsistent linear system.

[latex]\left[\begin{array}{ccc} 1 & 2 & 3\\ 4 & h & 1 \end{array}\right][/latex]