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Lecture 13

MATH125 Lecture 13: 13

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Department
Mathematics
Course Code
MATH125
Professor
Nikita

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Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ Gauss-Jordan elimination ▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ De▯nition. A matrix is in reduced echelon form if ▯▯ It is in row echelon form. ▯▯ The leading entry in each nonzero row is ▯ (called a leading 1). ▯▯ Each column containing a leading ▯ has zeroes everywhere else (below and above this leading 1). Example. ▯▯▯ ▯▯▯▯▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ 2 ▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ Example. ▯▯▯ ▯▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ 2 ▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ Example. ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ Example. ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯ ▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ Example. ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯ ▯ w ▯ x ▯ y ▯ ▯z ▯ ▯ ▯w ▯ ▯x ▯ y ▯ ▯z ▯ ▯ ▯ ▯w ▯ x ▯ y ▯ ▯▯ Solution. ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯ ▯ ▯ ▯▯ ▯▯ ▯ j ▯ ▯ ▯ ▯ ▯ ▯▯ j ▯ ▯: ▯ ▯ ▯ ▯ j ▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯ ▯▯ ▯▯▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯ ▯ j ▯ ▯ ▯ ▯ ▯ ▯▯ j ▯ ▯ : ▯ ▯ ▯ ▯ j ▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯ { w ▯ x ▯ z ▯ ▯ y ▯ z ▯ ▯ ▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ w ▯ ▯ ▯ x ▯ z▯▯▯ y ▯ ▯ ▯ z: ▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ x ▯ s ▯▯▯ z ▯ t ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ w ▯ ▯ s ▯ t ▯ ▯ ▯▯ ▯ x ▯ ▯ s ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯ ▯ s▯ ▯▯ t▯ ▯ : ▯ y ▯ ▯ ▯ ▯ t ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯▯ z t ▯ ▯ ▯ Example. ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ x ▯ ▯y ▯ z ▯ ▯ ▯▯▯ ▯x ▯ ▯y ▯ z ▯ ▯▯ Solution. ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯ ▯▯;▯;▯▯▯ ▯▯▯ ▯▯;▯;▯▯ ▯ ▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ax ▯▯by ▯ cz ▯ d ▯▯ ▯▯▯ ▯▯▯▯▯▯ a ▯ ▯ b : c ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ R ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ { x ▯ ▯y ▯ z ▯ ▯ ▯x ▯ ▯y ▯ z ▯ ▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ [ ] ▯ ▯ ▯▯ j ▯ ▯ ▯ ▯ j ▯ ▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ [ ] ▯ ▯ ▯▯ j ▯ ▯ ▯▯ ▯ j ▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯ [ ]
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