Dive into the second half of row reduction in linear algebra through this 49-minute lecture. Explore how parameters are introduced for non-leading columns in augmented matrices when solving linear equation systems. Learn to apply this technique to express vectors as linear combinations of others in both two and three dimensions. Work through multiple examples, including parallel lines in a plane, intersecting planes in 3D space, and writing vectors as linear combinations. Understand the concept of fully reduced row echelon form and its importance in solving systems. Practice with exercises to reinforce your understanding of these fundamental linear algebra concepts.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ row reduction using example; get all leading entry values to 1; fully reduced row echelon form; glorified highschool algebra done very systematically;
pg 2: @ example 2 2 parallel lines in a plane; no solution; example 3 3 equations, 3 variables; back substitution;
pg 3: @ definition: Fully reduced row echelon form; examples;
pg 4: @ examples; obtaining fully reduced row echelon matrices;
pg 5: @ Parameters; example a line; parametric solution;
pg 6: @ example a plane; parametric form for a plane: point on plane and 2 direction vectors;
pg 7: @ example intersection of 2 planes in 3D;
pg 8: @ why introduction of parameters works;
pg 9: @ problem1: writing a vector as a linear combination of 3 other vectors; important point: a system can be thought of in different terms;
pg 10: @ problem1 continued;
pg 11: @ problem2: writing 2d vector as a linear combination of 3 other 2d vectors;
pg 12: @ problem2 continued;
pg 13: @ exercise 14.1;
pg 14: @ exercise 14.2;
pg 15: @ exercise 14.3; THANKS to EmptySpaceEnterprise
Taught by
Insights into Mathematics
