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