Chapter 2. The Solution of linea Systems AX=B 2.1 Introduction to vectors and Matrices 2.2 Properties of Vectors and Matrices 2.3 Upper-triangular Linear Systems
Chapter 2. The Solution of Linear Systems AX=B 2.1 Introduction to Vectors and Matrices 2.2 Properties of Vectors and Matrices 2.3 Upper-triangular Linear Systems
Definition 2. 2. An N X N matrix A=aii is called upper triangular provided that the elements satisfy ai;=0 whenever i>j. The N N matrix A=aii is called lower triangular provided that ai =0 whenever i< j a111+a12x2+a133+…+a1N1xN-1+a1NxN=b1 a22+a23x3+…+a2N-1xN-1+a2NxN=b2 a33℃3+…+a3N-1xN-1+a3NxN=b3 aN-IN_IN-1+aN-INUN= bN-1 aNNEN E ON
Theorem 3.5 (Back Substitution). Suppose that AX=B is an upper- triangular system with the form given in(1). If k≠0fork=1,2,…,N, n there exists a unique solution to
Constructive Proof. The solution is easy to find. The last equation involves only N, So we solve it first N aNN Now N is known and it can be used in the next-to-last equation N-1 N-1N-1 NOW N and IN-I are used to find IN-2 N-2- aN-2N-1N-1-aN-2NN N-2 N-2N-2 Once the value N, N-1,., k+1 are known, the general step is N 21=k+1kj k for k=N-1.N-2 2.6 The uniqueness of the solution is easy to see. The Nth equation implies that bN/aNN is the only possible value of aN. Then finite induction is used to establish that N-1,N-2,…, are unique
Example 3. 12. Use back substitution to solve the linear system 4x1-x2+2c3+3x4=20 2x2+7x3-44=-7 63+5x4=-4 34=-6
Solving for 4 in the last equation yields Using 2 in the third equation, we obtain 6-5(2) 3 Now 3=-1 and 4=2 are used to find 2 in the second equation 7-7(-1)+4(2) Finally, c1 is obtained using the first equation 20+1(-4)-2(-1)-3(2) 3. 4 The condition that akk #0 is essential because equation(2.6) involves division by akk. If this requirement is not fulfilled, either no solution exists or infinitely many solutions exist
Example 3. 13. Show that there is no solution to the linear system I 4x1-x2+2x3+34=20 0℃2+703-4x4=-7 63+54=-4 3 2 4
Example 3. 14. Show that there are infinitely many solutions to 4x1-2+2x3+34=20 0x2+7x3-0x4=-7 6x3+54=-4 3x℃
Theorem 3.6. If the N X N matrix A=ai] is either upper or lower triangular, then N det(a)=a11a22 am=∏a i=1
2.4 Gaussian Elimination and Pivoting
2.4 Gaussian Elimination and Pivoting