Final Answer:
The triangular system solved by back substitution implies that the error is zero (Option a).
Step-by-step explanation:
Back substitution is a method used to solve triangular systems of linear equations, particularly those in upper triangular form. In this context, Theorem 17.1 indicates that the solution obtained by back substitution is exact, meaning it satisfies the system of equations without any error. Therefore, the correct answer is (a) "The error is zero." This implies that the back substitution process yields a solution that precisely satisfies the given triangular system of equations.
The mathematical rationale behind this conclusion lies in the nature of triangular systems. Back substitution starts from the bottom of the triangular system and works its way up, determining the values of variables one by one. Since each step is based on the previously determined exact values, the final solution is a precise solution without any error. This aligns with the zero error interpretation and underscores the reliability of back substitution for solving triangular systems.
In summary, (Option a)the back substitution method applied to a triangular system ensures that the obtained solution is exact, resulting in a zero error according to Theorem 17.1. Understanding the principles behind back substitution and its application to specific types of linear systems is fundamental for accurate and reliable mathematical solutions.