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# 1: Systems of Equations

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• 1.1: Geometry
• 1.2: Elementary Operations
We have taken an in depth look at graphical representations of systems of equations, as well as how to find possible solutions graphically. Our attention now turns to working with systems algebraically.
• 1.3: 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. First, we will represent a linear system with an augmented matrix. A matrix is simply a rectangular array of numbers. The size or dimension of a matrix is defined as m×n where m is the number of rows and n is the number of columns.
• 1.4: Uniqueness of the Reduced Row-Echelon Form
As we have seen in earlier sections, we know that every matrix can be brought into reduced row-echelon form by a sequence of elementary row operations. Here we will prove that the resulting matrix is unique; in other words, the resulting matrix in reduced row-echelon does not depend upon the particular sequence of elementary row operations or the order in which they were performed.
• 1.5: Rank and Homogeneous Systems
• 1.6: Balancing Chemical Reactions
The tools of linear algebra can also be used in the subject area of Chemistry, specifically for balancing chemical reactions.
• 1.7: Dimensionless Variables
This section shows how solving systems of equations can be used to determine appropriate dimensionless variables. It is only an introduction to this topic and considers a specific example of a simple airplane wing shown below. We assume for simplicity that it is a flat plane at an angle to the wind which is blowing against it with speed V as shown.
• 1.8: An Application to Resistor Networks
The tools of linear algebra can be used to study the application of resistor networks.
• 1.9: Exercises

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)