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