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1: Systems of Linear Equations

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    58777

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    A standard treatment of gaussian elimination is given. The rank of a matrix is introduced via the row-echelon form, and solutions to a homogeneous system are presented as linear combinations of basic solutions. Applications to network flows, electrical networks, and chemical reactions are provided.

    • 1.1: Solutions and Elementary Operations
      This page covers the fundamentals of linear algebra, focusing on solving systems of linear equations through various methods. It explains the nature of linear equations and their solutions, which can be consistent or inconsistent. The use of augmented matrices and elementary row operations to simplify and solve these systems is detailed, emphasizing the preservation of solutions through algebraic manipulation.
    • 1.2: Gaussian Elimination
      This page covers methods for solving systems of linear equations, focusing on row operations to achieve row-echelon and reduced row-echelon forms. It explains Gaussian elimination, the roles of leading and free variables, and provides examples illustrating these concepts. The uniqueness of the reduced row-echelon form is emphasized alongside the rank of matrices, which determines solution types: no solutions, unique solutions, or infinitely many solutions.
    • 1.3: Homogeneous Equations
      This page explores homogeneous systems of equations, explaining that they have all constant terms equal to zero, with the trivial solution being all variables equal to zero. It highlights the existence of nontrivial solutions when there are more variables than equations and discusses the relevance of linear combinations of vectors in this context.
    • 1.4: An Application to Network Flow
      This page explores the analysis of flow networks, including irrigation systems and traffic networks, emphasizing the Junction Rule where inflows equal outflows. It provides an example of a one-way street network, illustrating the derivation of flow equations at intersections. The text highlights the necessity for positive flow values and discusses practical constraints that may affect these flows.
    • 1.5: An Application to Electrical Networks
      This page covers the analysis of electrical networks, emphasizing the use of Ohm's Law (V = RI) and Kirchhoff’s Laws for calculating current. It defines resistors in ohms and voltage sources in volts, explains Kirchhoff's Junction Rule (current conservation at junctions), and the Circuit Rule (sum of voltage changes in a circuit). An example is provided to demonstrate the practical application of these principles for determining unknown currents.
    • 1.6: An Application to Chemical Reactions
      This page explains chemical reactions as the combination of molecules to create new substances, such as the formation of water from hydrogen and oxygen. It emphasizes the importance of balancing reactions, ensuring equal numbers of each type of atom on both sides. The process is illustrated using the example of burning octane, which involves solving equations for the molecule counts and requires positive integer coefficients, thus linking the concept to linear equations.
    • 1.E: Supplementary Exercises for Chapter 1
      This page addresses systems of linear equations, examining conditions for different solution outcomes (none, one, or infinitely many). It highlights that two equations in three variables cannot yield a unique solution if they are parallel or coincident. The content includes discussions on reduced row-echelon forms in matrices, practical problem-solving examples using ticket sales scenarios, and techniques for transforming nonlinear equations into linear ones.

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    This page titled 1: Systems of Linear Equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by W. Keith Nicholson via source content that was edited to the style and standards of the LibreTexts platform.