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Running head: GAUSSIAN ELIMINATION

Gaussian Elimination and its History Chelsea Casady Liberty University MATH 400

1

GAUSSIAN ELMINATION

2

Solving a system of linear equations can be completed using different methods: substitution, elimination, row reduction in matrices, and more. Elimination can be used to eliminate certain variables one by one until there is one unknown variable. When we solve for that one unknown, we can solve for the other unknowns. Substitution is used by isolating one variable and substituting that equation in to another equation to solve for the remaining variable; however, the method of substitution can be very tedious with more and more unknown variables. The method of Gaussian elimination can be used for a system of linear equations with many unknown variables. This method requires the use of matrices and matrix operations. According to Gareth Williams, author of Linear Algebra with Applications, “A matrix is a rectangular array of numbers” (2014). When we represent a system of equations in matrix form, we take out the variables and leave the coefficients and place them in rectangular form. For instance 2𝑥 + 3𝑦 + 𝑧 = 4 𝑥 + 3𝑦 + 3𝑧 = −4 𝑥+𝑦+𝑧 =0

can be written in matrix form as

2 1 1

3 3 1

1 3 1

4 -4 0

The method of Gaussian elimination and matrix operations to solve systems of linear equations stems from the Nine Chapters of Mathematical Art. This book was used in China when the Han dynasty began after the Qin dynasty was overthrown around 210 BCE. The Nine Chapters book was used for reference in the education system as it contained problems and answers of different mathematical topics. Joseph W. Dauben writes in The Mathematics of Egypt, Mesopotamia, China, India, and Islam, that “[I]t is a rich guide to the mathematics of ancient China because it comprises the entire range of then-known mathematics, from basic operations for computing with fractions to difficult problems involving matrix methods” (2007). The education system in China followed this book’s method for many centuries later as China strived to become superior in their mathematics. In chapter 8 of the book, a method of solving

GAUSSIAN ELMINATION

3

systems of linear equations is introduced and explained. This method is used to solve equations with three unknown variables. This method is different than the Babylonians method of solving for two unknown variables which was called false position. The Babylonians would pick arbitrary wrong values for the variables and manipulate the variables to find the correct values. For example, if we were 1

to solve this system of linear equations:

2

1

𝑥 − 5 𝑦 = 29 and x + y = 100, we would solve by

arbitrarily picking that x = y = 50. Plug those values into the first equation and we get: 1 1 (50) − (50) = 15 2 5 We want this to equal 29, so we must evaluate using the difference of 29 - 15 = 14. The Babylonians knew that if x increases, then y has to decrease. If we increase x by one unit, and 1

decrease the y by one unit, we add 2 +

1

= 5

7

7

. Using this new fraction, we must solve 10 𝑠 = 14 10

for s and we get s = 20. When we put this all together, we would get x = 50 + s = 70 and y = 50 – s = 30. The modern way of solving for each variable would be to substitute 100 – y in for x in the first equation and solve for y. This method is called substitution. The Nine Chapters worked with three linear equations with three unknown variables. Victor Katz gives a full example of problem one from the book in History of Mathematics (2004). He goes over problem 1 from Nine Chapters which says: There are three classes of grain, of which three bundles of the first class, two of the second, and one of the third make 39 measures. Two of the first, three of the second, and one of the third of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of grain are contained in one bundle of each class? (p. 126-127) In order to solve this system, it must be translated into the following linear equations:

GAUSSIAN ELMINATION

3x + 2y + z = 39 2x + 3y + z = 34 x + 2y + 3z = 26

4

They were then translated into the following array

1 2 3 26

2 3 1 34

3 2 1 39

The next step is to manipulate the second column so that the first number in the middle column is a zero. They did this by finding a common multiple of the 2 and 3 (the first numbers of the second and third columns) which is 6. They multiplied the second column by 3 to obtain 6 as the first number and then subtracted the third column multiplied by 2. The new second column becomes: 0

5

1

24. The next step is to go through the same process with the first and third

columns in order to make the first number of the first column a zero. Multiply the first column by 3 and subtract the third column which leaves the following first column: 0

4

8

39. We

now have the array: 0 4 8 39

0 5 1 24

3 2 1 39

The last manipulation step is to translate the first column using the second column so that the second number of the first column is a zero. Multiply the first column by 5 and subtract the second column multiplied 4. The author of the Nine Chapters rewrote the final array back into a system of linear equations. 0 0 36 99

0 5 1 24

3 2 1 39

3x + 2y + z = 39 5y + z = 24 36z = 99

Then the author solved for z and used back substitution to solve for y by plugging in z to the second equation. Then plug in y and z into the first equation to find x. From the article, History of Gaussian Elimination, “This ancient concept [from Nine Chapters] is the essence of what is now generally known as the method of Gaussian elimination”

GAUSSIAN ELMINATION

5

(1982). Gaussian elimination is a method used to eliminate variables in a system of linear equations using matrices. The major difference in the method used by the Chinese and Gauss was that equations are written in rows for Gaussian elimination rather than columns (History of Gaussian Elimination, 1982). In addition to the example found in the Nine Chapters, Isaac Newton studied systems of linear equations and found that in order to solve simultaneous equations, you must remove one variable from an equation by using the other equation (Joseph F. Grcar). This specifically relates to a system of two equations with two unknown. In modern times, this method would be called elimination; however, Newton called it “extermino” which translates in English as “exterminate” (Grcar). With knowledge of other fields besides mathematics and with previous mathematical works by the Chinese and other mathematicians like Newton, Carl Friedrich Gauss developed a method around 1810 for solving systems with more than two unknown variables which is now known as Gaussian elimination. Williams, in Linear Algebra with Applications describes three row operations that can be used to manipulate a matrix to find a solution of the system: “1. Interchange two rows of a matrix. 2. Multiply the elements of a row by a nonzero constant. 3. Add a multiple of the elements of one row to the corresponding elements of another row” (2014). In Gaussian elimination, we must use these operations to reduce the matrix into echelon form. A matrix is in echelon form if the first nonzero term in a row is a 1 (this is called a leading 1) and all numbers underneath a leading 1 are zeros (Williams, 2014). We can use the three elementary row operations listed to solve the system: x – 2y +4z = 12 2x – y + 5z = 18 -x + 3y -3z = -8

We start by translating into a matrix

1 2 -1

-2 -1 3

4 5 -3

12 18 -8

GAUSSIAN ELMINATION

6

Since the first nonzero number of the first row is a 1, we want all of the numbers underneath to be 0. We make the first number of the second row a 0 by using the third row operation and multiply the second row by -2 and add the second row. We make the first number of the third row a zero by adding the first row and the third row.

1 2 -1

-2 -1 3

4 5 -3

12 18 -8

1 0 0

R2 + (-2)R1 R3 +R1

-2 3 1

4 -3 1

12 -6 4

Next we must make the first nonzero term of the second row a leading 1. We do this by using the second row operation and multiply the second row by (

1 3

) R2

1 0 0

-2 1 1

4 -1 1

1 3

.

12 -2 4

Now that the first nonzero term of the second row is a leading 1, we want to manipulate the third row so that the number under the leading 1 of the second row is a zero. Multiply -1 by the second row and add the third row.

R3 + (-1)R2

1 0 0

-2 1 0

4 -1 2

12 -2 6

The last manipulation is to make the first nonzero term in the third row a zero. Multiply the third 1

row by , and we will have a matrix that satisfies the definition of echelon form. Our ending 2

matrix can be written as the following:

1 0 0

-2 1 0

4 -1 1

12 -2 3

Which can be written as the following equations: x – 2y + 4z = 12 y – z = -2 z=3

We use back substitution just like how the Chinese used to solve for the rest of the variables. x=2,y=1,z=3

GAUSSIAN ELMINATION

7

Originally when Gauss formed this method, matrices had not been developed yet, so Gauss manipulated the equations in rectangular form, but the rectangular form was not called a matrix until later in the nineteenth century (Grcar). The method of Gaussian elimination is used today in classrooms in the secondary and collegiate levels. This method of matrix transformations has evolved over the years as more methods have been developed using the same techniques. A specific evolved method that came from this elimination is the Gauss-Jordan Elimination which eliminates all numbers in the matrix except for the leading 1s using the same elementary row operations. In addition, with the advancement in technology, matrix algebra and transformations are programmed into calculators and computers, so by plugging in the numbers of a matrix, a system of linear equations can be solved in seconds.

GAUSSIAN ELMINATION

8 References

Grcar, J. (n.d.). Mathematicians of Gaussian Elimination. Retrieved April 15, 2015, from http://www.ams.org/notices/201106/rtx110600782p.pdf History of Gaussian Elimination. (1982). Retrieved April 9, 2015, from http://meyer.math.ncsu.edu/Meyer/PS_Files/GaussianEliminationHistory.pdf Imhausen, A., Robson, E., Dauben, J., Plofker, K., & Berggren, L. (2007). Chinese Mathematics. In V. Katz (Ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam (pp. 227-230). Princeton, New Jersey: Princeton University Press. Katz, V. (2004). A History of Mathematics: Brief Edition. New York: Pearson Education. Williams, G. (2014). Linear Algebra with Applications (8th ed.). Burlington, MA: Jones & Bartlett Learning.

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Gaussian Elimination and its History Chelsea Casady Liberty University MATH 400

1

GAUSSIAN ELMINATION

2

Solving a system of linear equations can be completed using different methods: substitution, elimination, row reduction in matrices, and more. Elimination can be used to eliminate certain variables one by one until there is one unknown variable. When we solve for that one unknown, we can solve for the other unknowns. Substitution is used by isolating one variable and substituting that equation in to another equation to solve for the remaining variable; however, the method of substitution can be very tedious with more and more unknown variables. The method of Gaussian elimination can be used for a system of linear equations with many unknown variables. This method requires the use of matrices and matrix operations. According to Gareth Williams, author of Linear Algebra with Applications, “A matrix is a rectangular array of numbers” (2014). When we represent a system of equations in matrix form, we take out the variables and leave the coefficients and place them in rectangular form. For instance 2𝑥 + 3𝑦 + 𝑧 = 4 𝑥 + 3𝑦 + 3𝑧 = −4 𝑥+𝑦+𝑧 =0

can be written in matrix form as

2 1 1

3 3 1

1 3 1

4 -4 0

The method of Gaussian elimination and matrix operations to solve systems of linear equations stems from the Nine Chapters of Mathematical Art. This book was used in China when the Han dynasty began after the Qin dynasty was overthrown around 210 BCE. The Nine Chapters book was used for reference in the education system as it contained problems and answers of different mathematical topics. Joseph W. Dauben writes in The Mathematics of Egypt, Mesopotamia, China, India, and Islam, that “[I]t is a rich guide to the mathematics of ancient China because it comprises the entire range of then-known mathematics, from basic operations for computing with fractions to difficult problems involving matrix methods” (2007). The education system in China followed this book’s method for many centuries later as China strived to become superior in their mathematics. In chapter 8 of the book, a method of solving

GAUSSIAN ELMINATION

3

systems of linear equations is introduced and explained. This method is used to solve equations with three unknown variables. This method is different than the Babylonians method of solving for two unknown variables which was called false position. The Babylonians would pick arbitrary wrong values for the variables and manipulate the variables to find the correct values. For example, if we were 1

to solve this system of linear equations:

2

1

𝑥 − 5 𝑦 = 29 and x + y = 100, we would solve by

arbitrarily picking that x = y = 50. Plug those values into the first equation and we get: 1 1 (50) − (50) = 15 2 5 We want this to equal 29, so we must evaluate using the difference of 29 - 15 = 14. The Babylonians knew that if x increases, then y has to decrease. If we increase x by one unit, and 1

decrease the y by one unit, we add 2 +

1

= 5

7

7

. Using this new fraction, we must solve 10 𝑠 = 14 10

for s and we get s = 20. When we put this all together, we would get x = 50 + s = 70 and y = 50 – s = 30. The modern way of solving for each variable would be to substitute 100 – y in for x in the first equation and solve for y. This method is called substitution. The Nine Chapters worked with three linear equations with three unknown variables. Victor Katz gives a full example of problem one from the book in History of Mathematics (2004). He goes over problem 1 from Nine Chapters which says: There are three classes of grain, of which three bundles of the first class, two of the second, and one of the third make 39 measures. Two of the first, three of the second, and one of the third of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of grain are contained in one bundle of each class? (p. 126-127) In order to solve this system, it must be translated into the following linear equations:

GAUSSIAN ELMINATION

3x + 2y + z = 39 2x + 3y + z = 34 x + 2y + 3z = 26

4

They were then translated into the following array

1 2 3 26

2 3 1 34

3 2 1 39

The next step is to manipulate the second column so that the first number in the middle column is a zero. They did this by finding a common multiple of the 2 and 3 (the first numbers of the second and third columns) which is 6. They multiplied the second column by 3 to obtain 6 as the first number and then subtracted the third column multiplied by 2. The new second column becomes: 0

5

1

24. The next step is to go through the same process with the first and third

columns in order to make the first number of the first column a zero. Multiply the first column by 3 and subtract the third column which leaves the following first column: 0

4

8

39. We

now have the array: 0 4 8 39

0 5 1 24

3 2 1 39

The last manipulation step is to translate the first column using the second column so that the second number of the first column is a zero. Multiply the first column by 5 and subtract the second column multiplied 4. The author of the Nine Chapters rewrote the final array back into a system of linear equations. 0 0 36 99

0 5 1 24

3 2 1 39

3x + 2y + z = 39 5y + z = 24 36z = 99

Then the author solved for z and used back substitution to solve for y by plugging in z to the second equation. Then plug in y and z into the first equation to find x. From the article, History of Gaussian Elimination, “This ancient concept [from Nine Chapters] is the essence of what is now generally known as the method of Gaussian elimination”

GAUSSIAN ELMINATION

5

(1982). Gaussian elimination is a method used to eliminate variables in a system of linear equations using matrices. The major difference in the method used by the Chinese and Gauss was that equations are written in rows for Gaussian elimination rather than columns (History of Gaussian Elimination, 1982). In addition to the example found in the Nine Chapters, Isaac Newton studied systems of linear equations and found that in order to solve simultaneous equations, you must remove one variable from an equation by using the other equation (Joseph F. Grcar). This specifically relates to a system of two equations with two unknown. In modern times, this method would be called elimination; however, Newton called it “extermino” which translates in English as “exterminate” (Grcar). With knowledge of other fields besides mathematics and with previous mathematical works by the Chinese and other mathematicians like Newton, Carl Friedrich Gauss developed a method around 1810 for solving systems with more than two unknown variables which is now known as Gaussian elimination. Williams, in Linear Algebra with Applications describes three row operations that can be used to manipulate a matrix to find a solution of the system: “1. Interchange two rows of a matrix. 2. Multiply the elements of a row by a nonzero constant. 3. Add a multiple of the elements of one row to the corresponding elements of another row” (2014). In Gaussian elimination, we must use these operations to reduce the matrix into echelon form. A matrix is in echelon form if the first nonzero term in a row is a 1 (this is called a leading 1) and all numbers underneath a leading 1 are zeros (Williams, 2014). We can use the three elementary row operations listed to solve the system: x – 2y +4z = 12 2x – y + 5z = 18 -x + 3y -3z = -8

We start by translating into a matrix

1 2 -1

-2 -1 3

4 5 -3

12 18 -8

GAUSSIAN ELMINATION

6

Since the first nonzero number of the first row is a 1, we want all of the numbers underneath to be 0. We make the first number of the second row a 0 by using the third row operation and multiply the second row by -2 and add the second row. We make the first number of the third row a zero by adding the first row and the third row.

1 2 -1

-2 -1 3

4 5 -3

12 18 -8

1 0 0

R2 + (-2)R1 R3 +R1

-2 3 1

4 -3 1

12 -6 4

Next we must make the first nonzero term of the second row a leading 1. We do this by using the second row operation and multiply the second row by (

1 3

) R2

1 0 0

-2 1 1

4 -1 1

1 3

.

12 -2 4

Now that the first nonzero term of the second row is a leading 1, we want to manipulate the third row so that the number under the leading 1 of the second row is a zero. Multiply -1 by the second row and add the third row.

R3 + (-1)R2

1 0 0

-2 1 0

4 -1 2

12 -2 6

The last manipulation is to make the first nonzero term in the third row a zero. Multiply the third 1

row by , and we will have a matrix that satisfies the definition of echelon form. Our ending 2

matrix can be written as the following:

1 0 0

-2 1 0

4 -1 1

12 -2 3

Which can be written as the following equations: x – 2y + 4z = 12 y – z = -2 z=3

We use back substitution just like how the Chinese used to solve for the rest of the variables. x=2,y=1,z=3

GAUSSIAN ELMINATION

7

Originally when Gauss formed this method, matrices had not been developed yet, so Gauss manipulated the equations in rectangular form, but the rectangular form was not called a matrix until later in the nineteenth century (Grcar). The method of Gaussian elimination is used today in classrooms in the secondary and collegiate levels. This method of matrix transformations has evolved over the years as more methods have been developed using the same techniques. A specific evolved method that came from this elimination is the Gauss-Jordan Elimination which eliminates all numbers in the matrix except for the leading 1s using the same elementary row operations. In addition, with the advancement in technology, matrix algebra and transformations are programmed into calculators and computers, so by plugging in the numbers of a matrix, a system of linear equations can be solved in seconds.

GAUSSIAN ELMINATION

8 References

Grcar, J. (n.d.). Mathematicians of Gaussian Elimination. Retrieved April 15, 2015, from http://www.ams.org/notices/201106/rtx110600782p.pdf History of Gaussian Elimination. (1982). Retrieved April 9, 2015, from http://meyer.math.ncsu.edu/Meyer/PS_Files/GaussianEliminationHistory.pdf Imhausen, A., Robson, E., Dauben, J., Plofker, K., & Berggren, L. (2007). Chinese Mathematics. In V. Katz (Ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam (pp. 227-230). Princeton, New Jersey: Princeton University Press. Katz, V. (2004). A History of Mathematics: Brief Edition. New York: Pearson Education. Williams, G. (2014). Linear Algebra with Applications (8th ed.). Burlington, MA: Jones & Bartlett Learning.

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