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Lecture 11

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University of Waterloo
MATH 136
Robert Sproule

Wednesday, January 29 − Lecture 11 : Homogeneous systems. Concepts: 1. homogeneous systems 2. trivial solution 11.1 Definition − Homogeneous system. A system of the form a11 1+ a 12 2... + a x1n n a21 1 a x22 2.. + a x 2n n : : a m1 1+ a m2 2... + a x mn n is called a homogeneous system. It can be represented more succinctly as [A | 0 ] where A is it coefficient matrix. Given a system of linear equations a 11 1a x 12 2. + a x =1n n 1 a 21 1a x 22 2. + a x =2n n 2 : : am1 1+ a m2 2... + a x mn n m represented as [A | b] we will say that the system [A | 0 ] is its “associated homogeneous system”. 11.1.1 Note − A homogeneous system always has at least one solution: the trivial ... solution x 1 = x2= = x n 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 mequations and n unknowns where m < n. Suppose we have transformed the associated augmented matrix [A m × n| 0] into a RREF matrix [A RREF | 0]. - Each the n columns of the matrix A RREF points to precisely one of the n variables. - It is possible that RREF has a row of zeros, but each row which contains non-zero entries must have a leading-1. So # basic variables = # leading-1’s ≤ m < # columns = n = # variables - So “# basic variables < # variables” implies there are free unknowns. The existence of at least one free variableguarantees that there are infinitely many solutions, as claimed. Question: If we have a non-homogeneous system where the number of rows m is less than the number of variables n are we guaranteed to have infinitely many solutions (just like homogeneous systems)? Answer: No. Note that if a homogeneous system with a number n of equations equal to the number n of unknowns has no row of zeros in its RREF then this system can only have the trivial solution. In fact its RREF matrix has only 1's on its diagonal and zeros as other entries. 11.2 Examples. We verify that for the system 3x1+ 5x 2 4x = 3 −3x 1 2x +24x = 03 6x 1 x −28x = 03 Row reducing to reduced row echelon form we get x and x as basic variablesand x 1 2 3 as a free variable and obtain the solution set and x = t( 4/3, 0, 1) . Note: It is important to be able to express the complete set of solutions to a system in the form of a vector equation. 11.2.1 Question: Are non-homogeneous systems with more equations than unknowns necessarily inconsistent? Answer: No; we may still end up with an RREF with rows of zeroes at the bottom which produces free unknowns. 11.2.2 Question: Does a homogeneous system with the same number of equations as unknowns always have only the trivial solutions? Answer: No. The RREF of the matrix may have rows of zeroes at the bottom. 11.3 Exercise question −For which values of a, b, c will the following system of equations be consistent? For those values, give the general solution and state whether or not it will be unique. x + z = a 2x + 6y + 4z = b 3y + z = c The system is consistent provided So if it is consistent we have or a line in 3-space. 11.4 Theorem − Suppose [A m × n| 0] is a homogeneous system of linear equations. n Suppose the rank of A is r. Then the complete set solution of [A m × n| 0] is a subset of ℝ which can be expressed in the form span{a , a , a 1 …,2a 3 n - r. Proof : A general idea of how the proof flows is given in class. 11.4.1 Corollary − Suppose p and p a1e any tw2 solutions of the nystem [A
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