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- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/11%3A_Appendix_A-_Linear_Algebra/11.03%3A_A.3-_EliminationWe start with the first column and we locate the pivot, in this case the first entry of the first column. \[\left[ \begin{array}{ccc|c} \fbox{2} & 2 & 2 & 2 \\ 1 & 1 & 3 & 5 \\ 1 & 4 & 1 & 10 \end{arr...We start with the first column and we locate the pivot, in this case the first entry of the first column. [2222113514110] We multiply the first row by 12. [1111113514110] We subtract the first row from the second and third row (two elementary operations). \[\left[ \begin{array}{c…
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.08%3A_The_Rank_TheoremThis page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space...This page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space) equals the number of columns. It includes examples demonstrating how different ranks and nullities influence solution options in linear equations, emphasizing the theorem's importance in understanding the relationship between solution freedom and system properties without direct calculations.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/Appendix_A%3A_Linear_Algebra/A.3%3A_EliminationThis page provides an in-depth explanation of solving linear systems of equations using matrices and techniques such as elimination and Gauss-Jordan elimination. It discusses transforming systems into...This page provides an in-depth explanation of solving linear systems of equations using matrices and techniques such as elimination and Gauss-Jordan elimination. It discusses transforming systems into matrix equations, performing row operations to achieve row echelon or reduced row echelon form, and distinguishing cases where solutions are unique, non-unique, or non-existent. Concepts like linear independence, rank, nullspace, and kernel are introduced alongside practical examples.
