1: Systems of Equations
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- 1.1: Systems of Linear Equations
- This page discusses methods for solving systems of equations with two or three variables, covering unique, infinite, and no solutions. It emphasizes the significance of graphical representations, intersections of lines and planes, and the complexities introduced by additional variables. The page addresses consistent versus inconsistent systems, homogeneous systems, and the application of elementary operations that preserve solution sets.
- 1.2: Gaussian Elimination
- The work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions, by representing a linear system as an augmented matrix and performing a systematic set of operations on it to reach a solution to the system.
- 1.3: Rank and Homogeneous Systems
- There is a special type of system which requires additional study. This type of system is called a homogeneous system of equations. Our focus in this section is to consider what types of solutions are possible for a homogeneous system of equations. We also define two important terms: linear combination and rank.
- 1.4: Balancing Chemical Reactions
- The tools of linear algebra can also be used in the subject area of Chemistry, specifically for balancing chemical reactions.
- 1.5: Applications to Physics
- The tools of linear algebra can be used to study the application of resistor networks and dimensionless variables.
- 1.6: Application to Polynomial Interpolation
- Given n points in R2, we solve a linear system to find an appropriate degree polynomial that passes through them.
- 1.7: Gauss-Seidel Method
- We consider an iterative method of solving a linear system in this section. Elimination methods, such as Gaussian elimination, are prone to large round-off errors for a large set of equations. Iterative methods, such as the Gauss-Seidel method, give the user control of the round-off error.
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)