A Guide to Gaussian Jordan Reduction: Topics in Linear Algebra

Introduction to Gaussian Jordan Reduction

This series regarding linear algebra has focused primarily on subject matters relating to facilitating understanding of linear systems. In our first article of this series, we elaborated in great depth on explicating the Gaussian method as it exists as a technique for solving linear systems. This article was immediately followed by another presentation addressing how solutions of linear systems can be described. Presently, we continue to explicate this thread of linear system computation in this discussion of Gaussian Jordan reduction.

Gaussian-Jordan Elimination

The Gaussian method exhibits a variety of methodologies which permit discernment of the solutions for linear systems. The Gaussian Jordan reduction method is one of such methods for solving linear systems.

In Gaussian Jordan reduction, the methodology begins as it always does by manipulating the linear system into echelon form. Recall that echelon form is a structural phenomena of the linear system wherein the leading variable of a subsequent equation is to the right of the leading variable of the preceding equation. If you have some apprehensions towards this concept, consider checking out our article which discusses transformation to echelon form.

Once the linear system has been manipulated into echelon form, we modify the system further by converting the leading variables for each equation to a one. Once in echelon form and leading variables are one, we undertake a series of computations that converts all other entries to zero, such that the only non-zero entry in the equation is the leading variable, which is one. This cumulative procedure represents Gaussian Jordan reduction. In this form, the solutions of the linear system are already explicitly clear.

We may deliver an unambiguous definition for the reduced echelon form achieved by Gaussian Jordan reduction. A linear system is in reduced echelon form if the system is in echelon form, the leading variable is one, and all other entries possess a value of zero.

Example of Gaussian-Jordan Elimination

We now utilize this opportunity to provide an example of how to use Gaussian Jordan reduction to derive the solutions. Consider the following system:

\begin{pmatrix} 2 & 1 & |\enspace7 \\ 4 & -2 & |\enspace6 \end{pmatrix}
Steps

Firstly, this initial linear system is not in echelon form. Therefore, our first priority is to manipulate this system into echelon form. As we know, the principal goal of this is so that the leading variable of the second equation is to the right of the leading variable of the first equation. We can achieve this by subtracting two times the first equation from the second equation, which procures a system of the form:

\begin{pmatrix} 2 & 1 & |\enspace \enspace \enspace7 \\ 0 & -4 & |\enspace-8 \end{pmatrix}

We can see from the system above that it exhibits the features of echelon form (leading variable in equation two is to the right of leading variable of equation 1). Now that the system adheres to echelon form, we now have to achieve the next goal of making the leading variables one’s. We may achieve this by multiplying equation one by (1/2) and equation two by (-1/4). When we convey these computations, they produce the system that appears as follows:

\begin{pmatrix} 1 & \frac{1}{2} & |\enspace \frac{7}{2} \\ 0 & 1 & |\enspace2 \end{pmatrix}

Now, all we must do is make sure that all entries other than the leading variables are zero. We must do this by applying the bottom equation in some way to the top equation. Thus, we can add negative one half of equation two to the equation one. When we do this, we obtain the following system:

\begin{pmatrix} 1 & 0 & |\enspace \frac{5}{2} \\ 0 & 1 & |\enspace2 \end{pmatrix}

Because the leading variables are themselves one, we know that the value of ‘x’ is 5/2 and the value of ‘y’ is 2. This is the solution to the system.

Trade-Offs of Gaussian Jordan Reduction

One of the drawbacks of utilizing the Gaussian Jordan reduction method is that it requires extra arithmetic for computation. However, applying this methodology permits several essential insights. Firstly, as previously asserted, the use of the reduced echelon form is the fact that because the leading variables are ‘1’, the system’s solution appears apparent. Furthermore, the size of the solution set also seems clear.

Lemmas of Reduced Echelon Form

Lemma #1: Elementary Row Operations Are Reversible

Any computational method that we apply to the row of a linear system can be reversed by an inverse computation. For example, if we multiply a row by two, then it may be reversed by multiplying one-half (or dividing by two).

Lemma #2: The Coinage ‘Reduces To’ Is a Statement of Equivalence

When we apply a method to a particular row by manipulation with another row, we often say that the row ‘reduces to’ the subsequent row. This is true providing three conditions are met. Firstly, the row satisfies the rule of reflexivity, wherein the matrix can reduce to itself. Secondly, the row adheres to the rule of transitivity. By this clause, if row A reduces to row B by computation, and row B reduces to row C by another computation, then A may be said to reduce directly or indirectly to C. Finally, the row also adheres to the law of symmetry, such that if row A reduces to row B, then row B may reduce to row A.

Lemma #3: If Two Matrices Are Reducible to Each Other, Then The Rows Are Row Equivalents

A given matrix or linear system may be thought of as belonging to a specific class. If the matrices are inter-reducible, then the matrices exist within the same class. However, if the matrices or rows aren’t inter-reducible, then these entities do not belong to the same class.

Summarizing Gaussian Jordan Reduction

From the features examined in this article, it is quite evident that Gaussian Jordan reduction belongs to a class of computation known as algorithms. Algorithms are procedural methodologies that exist as a series of steps which when followed lead to correct quantification of some parameter. Gaussian Jordan reduction certainly belongs to this classification.

As we previously saw in generalized Gaussian elimination, Gaussian Jordan reduction necessitates that a linear system must first be in echelon form. Once in echelon form, leading variables must be manipulated to exhibit one’s. Then, by modifying previous equations with subsequent equations, the solution to the system may be read directly. If these concepts still feel a bit confusing, do not worry. Later articles deal directly with applying examples of Gaussian Jordan reduction. However, the next subject of discussion deals with a discussion on the lemmas of linear combination. Thus, if you are in need of something to whet your appetite pertaining to Gaussian Jordan reduction, consider checking out this resource.

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