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1: Systems of Linear Equations

  • Page ID
    170419
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    • 1.1: Systems of Linear Equations
    • 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. 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.
    • 1.3: 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.
    • 1.4: Vectors
      We have been drawing points in Rⁿ as dots in the line, plane, space, etc. We can also draw them as arrows. Since we have two geometric interpretations in mind, we now discuss the relationship between the two points of view.
    • 1.5: Vector Equations and Spans
      The thing we really care about is solving systems of linear equations, not solving vector equations. The whole point of vector equations is that they give us a different, and more geometric, way of viewing systems of linear equations.
    • 1.6: Matrix Equations
      In this section we introduce a very concise way of writing a system of linear equations: Ax=b. Here A is a matrix and x,b are vectors (generally of different sizes).
    • 1.7: Solution Sets
      In this section we will study the geometry of the solution set of any matrix equation Ax=b.
    • 1.8: Applications of Linear Systems
      The tools of linear algebra can also be used in the subject area of Chemistry, specifically for balancing chemical reactions.
    • 1.9: Linear Independence
      Sometimes the span of a set of vectors is “smaller” than you expect from the number of vectors, as in the picture below. This means that (at least) one of the vectors is redundant: it can be removed without affecting the span. In the present section, we formalize this idea in the notion of linear independence.
    • 1.10: 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)


    This page titled 1: 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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