MATH-M 303 Lecture Notes - Lecture 5: Solution Set, Gaussian Elimination, Augmented Matrix
Document Summary
M303 section 1. 5 notes- solution sets of linear systems. Structure of solution sets does not have much variety; general system of algebraic equations would have much more variance. If there is one, system has infinitely many solutions. To find, row reduce to see if free variables appear; if all columns have pivots, zero solution is the only one: ex. Determine if the following homogeneous system has nontrivial solutions and give the solution set. System consistent; free, so system does have nontrivial solutions. Set of all scalar multiples of vector - its span. In general, solution set of is: ex. Give the solution set for . all linear combinations of. Non-homogeneous- where: solution set looks like a translate of solution set of corresponding homogeneous system, ex. Parametric vector form (general solution vector: when expanding, one vector will not have free variables in it. Homogeneous solutions are multiples of , translate by adding constant vector.
