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1: Solving Systems of Linear Equations

  • Page ID
    201831
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    • 1.1: Systems of Linear Equations
      This section discusses linear equations and systems, defining them and their classifications based on the existence and uniqueness of solutions. It covers geometric interpretations for two-variable systems (intersecting, parallel, or coinciding lines) and extends to three-variable systems (intersecting, non-intersecting, or coinciding planes). The text emphasizes the complexity that arises with additional variables, necessitating linear algebra for further analysis.
    • 1.2: Matrix notation and Row Reduction
      This section presents a comprehensive overview of solving systems of linear equations using augmented matrices and row operations. Key techniques include elementary row operations, achieving row echelon form and reduced row echelon form, and understanding the implications of basic and free variables. The document includes examples that demonstrate the application of these techniques, identifies cases of infinite solutions via parameters, and addresses practical mathematical problems.
    • 1.3: Rank and Homogeneous Systems
      This Section discusses homogeneous systems of linear equations, emphasizing the presence of the trivial solution and the conditions for nontrivial solutions when there are more variables than equations. It explains the rank of a matrix as a measure of linear independence, its derivation from pivot columns, and its importance in determining the nature of solutions.
    • 1.4: Chemistry Application- Balancing Chemical Reactions
      This section explains the use of augmented matrices and linear algebra for balancing chemical reactions, highlighting the Mass Balance Theory, which ensures constant elemental quantities. It includes detailed examples for balancing reactions involving tin and potassium with phosphorus, showcasing the method of creating and solving systems of equations.
    • 1.5: Applications to Physics and Resistor Networks
      This page explains the application of Kirchhoff’s Law and systems of equations for determining current in electrical circuits using linear algebra. It includes examples with resistors and voltage sources, leading to a system of equations for unknown currents. Additionally, the page discusses dimensionless variables in fluid dynamics, particularly in relation to aerodynamics, showcasing their significance in calculating lift and analyzing fluid behavior.
    • 1.6: Chapter Exercises
      This section offers an extra set of exercises on solving systems of linear equations, matrix row reduction techniques, and applications of system of linear equations in science and engineering.

    Thumbnail: A linear system in three variables determines a collection of planes. The intersection point is the solution. (CC BY-SA 4.0; Fred the Oyster via Wikipedia)


    This page titled 1: Solving Systems of Linear Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.

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