Remark 212 1 The first column of the augmented matrix corresponds to the coeffi

# Remark 212 1 the first column of the augmented matrix

• Essay
• 176

This preview shows page 25 - 27 out of 176 pages.

Remark 2.1.2. 1. The first column of the augmented matrix corresponds to the coeffi- cients of the variable x 1 . 2. In general, the j th column of the augmented matrix corresponds to the coefficients of the variable x j , for j = 1 , 2 , . . . , n . 3. The ( n + 1) th column of the augmented matrix consists of the vector b . 4. The i th row of the augmented matrix represents the i th equation for i = 1 , 2 , . . . , m . That is, for i = 1 , 2 , . . . , m and j = 1 , 2 , . . . , n, the entry a ij of the coefficient matrix A corresponds to the i th linear equation and the j th variable x j . Definition 2.1.3. For a system of linear equations A x = b , the system A x = 0 is called the associated homogeneous system . Definition 2.1.4 (Solution of a Linear System) . A solution of A x = b is a column vector y with entries y 1 , y 2 , . . . , y n such that the linear system (2.1.1) is satisfied by substituting y i in place of x i . The collection of all solutions is called the solution set of the system. That is, if y t = [ y 1 , y 2 , . . . , y n ] is a solution of the linear system A x = b then A y = b holds. For example, from Example 3.3a, we see that the vector y t = [1 , 1 , 1] is a solution of the system A x = b , where A = 1 1 1 1 4 2 4 10 1 , x t = [ x, y, z ] and b t = [3 , 7 , 13] . We now state a theorem about the solution set of a homogeneous system. The readers are advised to supply the proof. Theorem 2.1.5. Consider the homogeneous linear system A x = 0 . Then
26 CHAPTER 2. SYSTEM OF LINEAR EQUATIONS 1. The zero vector, 0 = (0 , . . . , 0) t , is always a solution, called the trivial solution. 2. Suppose x 1 , x 2 are two solutions of A x = 0 . Then k 1 x 1 + k 2 x 2 is also a solution of A x = 0 for any k 1 , k 2 R . Remark 2.1.6. 1. A non-zero solution of A x = 0 is called a non-trivial solution. 2. If A x = 0 has a non-trivial solution, say y negationslash = 0 then z = c y for every c R is also a solution. Thus, the existence of a non-trivial solution of A x = 0 is equivalent to having an infinite number of solutions for the system A x = 0 . 3. If u , v are two distinct solutions of A x = b then one has the following: (a) u v is a solution of the system A x = 0 . (b) Define x h = u v . Then x h is a solution of the homogeneous system A x = 0 . (c) That is, any two solutions of A x = b differ by a solution of the associated homogeneous system A x = 0 . (d) Or equivalently, the set of solutions of the system A x = b is of the form, { x 0 + x h } ; where x 0 is a particular solution of A x = b and x h is a solution of the associated homogeneous system A x = 0 . 2.1.1 A Solution Method Example 2.1.7. Solve the linear system y + z = 2 , 2 x + 3 z = 5 , x + y + z = 3 . Solution: In this case, the augmented matrix is 0 1 1 2 2 0 3 5 1 1 1 3 and the solution method proceeds along the following steps. 1. Interchange 1 st and 2 nd equation. 2 x + 3 z = 5 y + z = 2 x + y + z = 3 2 0 3 5 0 1 1 2 1 1 1 3 . 2. Replace 1 st equation by 1 st equation times 1 2 .
• • • 