Gaussian elimination

Gaussian elimination is a mathematical method used to solve systems of linear equations. Specifically, the method is used to solve systems with three or more variables. Gaussian elimination is based on addition and elimination, and the equations eventually wind up in row-echelon form. The method is named after German mathematician Carl Friedrich Gauss, who is regarded as one of the greatest mathematicians to ever live. Gauss also contributed to the fields of astronomy and physics.

Carl Friedrich Gauss

Gaussian elimination borrows its named from Carl Friedrich Gauss, a German mathematician. Born on April 30, 1777, in Brunswick, Germany, Gauss came from a poor, working-class family. Gauss’s father was a gardener and bricklayer. His father discouraged him from going to school, but his mother and uncle noticed that he was a brilliant child and encouraged him to obtain an education. Gauss soon developed impressive mathematics skills. These skills became evident when he was in elementary school. One day in arithmetic class, Gauss's teacher instructed the students in class to write on their slates the sum of all whole numbers from 1 to 100 and then turn in the slate. The exercise should have taken the students a great deal of time to complete. Gauss, however, completed the exercise within seconds. As it turned out, his answer was correct, and most of the other students’ answers were incorrect. Furthermore, Gauss’s slate contained just one number, 5050, which was the correct answer. He had quickly figured out a pattern for adding the whole numbers from 1 to 100: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on. There are fifty pairs of these numbers, each of which adds up to 101, which means that 50 × 101 = 5050.

In 1788, Gauss began attending the Gymnasium, a senior secondary school. There, he studied High German and Latin. Gauss went on to study at Caroline College (also called Brunswick Collegium Carolinum) and the University of Göttingen. Impressed by Gauss’s abilities, Charles (Karl) Wilhelm Ferdinand, Duke of Brunswick, financially supported Gauss throughout his college career. Gauss studied mathematics and even made an impressive discovery while in college. He determined that an individual could draw a seventeen-sided figure simply using a straightedge and a compass. Considered a significant contribution to the field of mathematics, Gauss’s discovery was published in his work Disquisitiones Arithmeticae. He also discovered Bode’s law and the law of quadratic reciprocity.

While at the University of Göttingen, Gauss submitted a proof on algebraic equations that became known as the fundamental theorem of algebra. He then became involved in the field of astronomy. Astronomers had discovered Ceres, which they thought was a new planet. They lost sight of Ceres, but Gauss helped them rediscover the so-called planet by calculating its position. This led to his development of a method for determining the orbits of new asteroids. Gauss eventually became the director of Göttingen’s observatory. He also became involved in differential geometry, publishing papers such as Disquisitiones generales circa superficies curva. His other works included Disquisitiones generales circa seriem infinitam, which included an introduction to the hypergeometric function, and Theoria combinationis observationum erroribus minimis obnoxiae and its supplement, which involved mathematical statistics, particularly the least squares method. Gauss also contributed to the field of physics, publishing papers such as Über ein neues allgemeines Grundgesetz der Mechanik and Principia generalia theoriae figurae fluidorum in statu aequilibrii, which were based on potential theory. Gauss also invented the heliotrope, an instrument that uses mirrors and a small telescope to reflect the Sun’s rays. He remained at Göttingen’s observatory until his death on February 23, 1855.

How Gaussian Elimination Is Performed

Gaussian elimination is used in linear algebra. It is performed on systems of linear equations, such as those that have three or more rows. The systems that are triangular in form are generally the easiest systems to solve using Gaussian elimination. This means that the lower rows of the system are shorter in length than the upper rows. The reason for this is because the lower rows contain fewer variables—such as x, y, and z—than the upper rows.

For example, a system has three rows with variables x, y, and z. The top row contains all three variables, the middle row contains only y and z, and the bottom row contains only z. This gives the system a triangular form. To perform Gaussian elimination, several steps must be followed. First, take the z-value from the equation in the bottom row and back-substitute it into the equation in the middle row. Then solve for y. Next, place z and y into the equation in the top row. Then solve for x. Because the system is triangular in form, it is relatively easy to solve using Gaussian elimination. For a system that is not triangular in form, Gaussian elimination can be used to reduce the system to this form.

Gauss-Jordan elimination is a method that is similar to Gaussian elimination. It is considered a more complete method than Gaussian elimination.

Bibliography

O’Connor, J J, and E F Robertson. “Johann Carl Friedrich Gauss.” School of Mathematics and Statistics, University of St Andrews, Scotland. Web. 18 Dec. 2014. http://www-history.mcs.st-and.ac.uk/Biographies/Gauss.html

Soualem, Nadir. “Gaussian Elimination.” Math-Linux.com. Math-Linux.com. 18 Jul. 2006. Web. 18 Dec. 2014. http://www.math-linux.com/mathematics/linear-systems/article/gaussian-elimination

Stapel, Elizabeth. “Systems of Linear Equations: Solving by Gaussian Elimination.” Purplemath. Purplemath. Web. 18 Dec. 2014. http://www.purplemath.com/modules/systlin6.htm

Weller, Karolee. “Carl Friedrich Gauss.” Wichita State University. Wichita State University. Web. 18 Dec. 2014. http://www.math.wichita.edu/history/men/gauss.html