MATH-M 303 Lecture Notes - Lecture 2: Elementary Matrix, Augmented Matrix, Row Echelon Form
Document Summary
M303 section 1. 2 notes- row reduction and echelon forms. Elementary row operations- to obtain equivalent systems. Ex: typically put tilde (~) somewhere near expressions of row ops to show that resulting matrix is similar equivalent to previous step elementary row operations. If (cid:1827) ~ (cid:1828), then (cid:1828) ~ (cid:1827) and (cid:1827) ~ (cid:1828) iff (cid:1828) ~ (cid:1827); relation of row equivalency is symmetric. If we perform a row op on any augmented matrix [(cid:1827)|] to get [(cid:1827) | ], any solution to original system is a solution of system corresponding to [(cid:1827) | ] (cid:886) (cid:882) (cid:883) (cid:885)(cid:882)| (cid:882)(cid:886) (cid:885)(cid:882)] (cid:2871) (cid:2871) (cid:883)(cid:882)(cid:2870) ~ [(cid:883) (cid:884) (cid:883) (cid:883) (cid:886) (cid:882) (cid:882) (cid:882) (cid:883)|(cid:882)(cid:886) (cid:883)] [(cid:883) (cid:884) (cid:883) (cid:2871) (cid:2869)(cid:2871)(cid:2868)(cid:2871) ~ (cid:882) (cid:883) (cid:886) (cid:882) (cid:882: can see (cid:2871)= (cid:883), but still need substitution; can continue to simplify by making zeroes above. Existence and uniqueness of solutions: ex. [(cid:1827)|]=[(cid:882) (cid:886) (cid:890) (cid:883)(cid:884)|(cid:890)(cid:883)(cid:883)] (cid:883) (cid:886) (cid:884) (cid:884) (cid:885)
