MATH136 Lecture Notes - Lecture 9: Hyperplane, Solution Set, Coefficient Matrix

• School: University of Waterloo
• Department: Mathematics
• Course: MATH136
• Professor: Alejandra Vicente Colmenares

Document Summary

Math 136 lecture 9 22 jan, 2018. Definition: a system of linear equations is a collection of linear equations in the same variables. Definition: a linear equation in variables (cid:2869), ,(cid:3041) is an equation that can be written in the form (cid:1853)(cid:2869)(cid:2869)+(cid:1853)(cid:2870)(cid:2870)+ +(cid:1853)(cid:3041)(cid:3041)=(cid:1854) (geometrically represents a hyperplane in (cid:3041)) The (cid:1853)(cid:3036) are called the coefficients, (cid:1854) is called the constant side or the "right-hand side". (cid:1853)(cid:2869)(cid:2869)(cid:2869)+(cid:1853)(cid:2869)(cid:2870)(cid:2870)+ +(cid:1853)(cid:2869)(cid:3041)(cid:3041)=(cid:1854)(cid:2869) (cid:1853)(cid:2870)(cid:2869)(cid:2869)+(cid:1853)(cid:2870)(cid:2870)(cid:2870)+ +(cid:1853)(cid:2870)(cid:3041)(cid:3041)=(cid:1854)(cid:2870) System of (cid:1865) equations in (cid:1866) variables. (cid:1853)(cid:3036)(cid:3037) the coefficient of (cid:3037) in the (cid:1861)-th equation. Theorem 2. 1. 1: let (cid:1871) and (cid:1872) both be solutions of a system of (cid:1865) linear equyations in (cid:1866) variables with (cid:1871) (cid:1872) , then for each (cid:1853) , =(cid:1871) +(cid:1853)(cid:4666)(cid:1872) (cid:1871) (cid:4667) is a unique solution of the system. (cid:1853)(cid:2869)(cid:2869)(cid:2869)+(cid:1853)(cid:2869)(cid:2870)(cid:2870)+ +(cid:1853)(cid:2869)(cid:3041)(cid:3041)=(cid:1854)(cid:2869) (cid:1853)(cid:2870)(cid:2869)(cid:2869)+(cid:1853)(cid:2870)(cid:2870)(cid:2870)+ +(cid:1853)(cid:2870)(cid:3041)(cid:3041)=(cid:1854)(cid:2870) Definition: a set of all solutions od a system of equations is called the solution set.