MATH136 Lecture Notes - General Idea, Coefficient Matrix, Solution Set

• School: University of Waterloo
• Department: Mathematics
• Course: MATH136
• Professor: Robert Andre

Document Summary

Wednesday, january 29 lecture 11 : homogeneous systems. It can be represented more succinctly as [a | 0 ] where a is it coefficient matrix. 11. 1. 1 note a homogeneous system always has at least one solution: the trivial solution x1 = x2 = = xn = 0 and so is always consistent. 11. 1. 2 claim : a homogeneous system with fewer equations than unknowns must have infinitely many (non-trivial) solutions. Proof of this claim: suppose a system has m equations and n unknowns where m < n. Suppose we have transformed the associated augmented matrix [am n | 0] into a. Each the n columns of the matrix arref points to precisely one of the n variables. It is possible that arref has a row of zeros, but each row which contains non-zero entries must have a leading-1. # basic variables = # leading-1"s m < # columns = n = # variables.