Chapter 2. The solution of linear Systems AX-B
Chapter 2. The Solution of Linear Systems AX=B
2.1 Introduction to vectors and Matrices
2.1 Introduction to Vectors and Matrices
2.2 Properties of Vectors and Matrices
2.2 Properties of Vectors and Matrices
2. 3 Upper-triangular Linear Systems
2.3 Upper-triangular Linear Systems
Definition 2.2. An N X N matrix A= a;i) is called upper triangular provided that the elements satisfy ai=0 whenever i>j. The N X N matrix A=ail is called lower triangular provided that ai=0 whenever i< j 1x1+a12x2+a133+…+a1N-1xN-1+a1NxN=b1 222+a23x3+…+a2N-1xN-1+a2NxN=b2 a33 3+.+a3N-1N-1+a3NCN=b3 aN___1+aN_INCN= bN-1 aNNeN=b
Theorem 3.5 (Back Substitution). Suppose that AX= B is an upper triangular system with the form given in(1). If k≠0rk=1,2,…,N, then there exists a unique solution to (1)
Constructive Proof. The solution is easy to find. The last equation involves only CN, So we solve it first (2.3 aNN NOw IN is known and it can be used in the next-to-last equation aN-IN CN N-1= (24 N-1N-1 Now N and IN-I are used to find N-2 ON-2-aN-2N-1CN-1-aN-2N CN 2 O nce the value N, N-1,., k+1 are known, the general step Is ali k fork=N-1,N-2,,1 akk e uniqueness of the solution is easy to see. The Nth equation implies that ON/aNn is the only possible value of CN. Then finite induction is used to establish that N-1,TN-2,……1 are unique
Example 3. 12. Use back substitution to solve the linear system 41-x2+23+34=20 2x2+7x3-4x4=-7 6x3+5x4=-4 3x4=-6
Solving for 4 in the last equation yields 4 Using a2 in the third equation, we obtain 6-5(2) Now 33=-1 and 4=2 are used to find 2 in the second equation 7-7(-1)+4(2) Finally, i is obtained using the first equation: 20+1(-4)-2(-1)-3(2) 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 41-x2+2x3+34=20 0x2+7x3-44=-7 63+54=-4 3 4
