Welcome to my math web site.  This web site has been designed by me in an attempt to promote mathematics at the college level.  On this site, I have posted solutions to various application problems that I have created and my written solutions to various proofs.  In part, I also created this site as a way to electronically compile information together regarding concepts that I have learned in different math courses throughout my career to help me with long term retention of the material.  I did not create this web page to fulfill any requirements for any courses that I have taken nor did I receive any external benefit of any sort for creating this page.  I created this web page only out of my pure enjoyment for the subject of mathematics and because I feel that the subject of math needs more attention at both the high school and college level.  This site is very young and I am in the process of updating it so please refer back to it routinely to catch the most recent updates. 

 

“The more math you learn, the more math you want to learn”

 

“The logical and critical thinking skills that are greatly gained from studying math can be carried over to benefit a person in many areas of life.” –After studying math for three years beyond high school, I came up with this quote.

 

Note:  All of the contents on this web site and any web site that this site links to have been produced by Ben Mueller.  Unless other wise stated, all application problems on this page have been created by Ben Mueller.  Also, the solutions to the various application problems and proofs on this page have been carefully written by Ben Mueller.  If you do notice an error in any written solution or proof on this page, please do not hesitate to email me at Muellerbt15@mail.uww.edu.  Finally, please do not reproduce or duplicate any text on this page without my written consent.  Thank you and enjoy the page.

 

Click here to read what others have said about this site.

 

Click here to read my personal math journal

My newest features:    

 

Conic Sections:  Geometric Application to Space

 

*****Essay regarding the benefits and dangers of the calculator (new)*****

 

*****Relating Math to the other Content Areas (New)*****

 

My Advice to Math Majors (take it for what it’s worth – probably nothing) 

Click here to read my personal math journal

Elementary Matrix and Linear Algebra: 

Abstract Algebra:

College Geometry:

Calculus and Analytic Geometry:

Number Systems

 

*****Click here to go to the Calculus on the Web (COW) website*****

 

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Regular real number multiplication is commutative.  This means that if r and s are real number, rs = sr.  For example, just about everyone knows that 2*3 = 3*2 = 6.  In other words, no matter what order we multiply the numbers, the product is always the same in real number multiplication.  In general, matrix multiplication is not commutative.  If A and B are matrices, it is not always true that A*B = B*A.  In fact, in most cases the product of matrix A and matrix B (if it exists) will result in a different matrix than the product of Matrix B and Matrix A. 

 

Note:  If A and B are matrices, A*B ≠ B*A (Matrix multiplication is not commutative).

 

It is sometimes interesting, however, to find all of the matrices for B where A*B = B*A given a matrix A. 

 

Application Problem (Problem created and solution derived by Ben Mueller)

 

Given the matrix A, find all matrices B such that A*B = B*A. 

 

Matrix A = 

 

Solution: 

 

Matrix B must be of the form:

                                                x,y,z and w are real number.

 

We can solve for our matrix B by finding solutions for x,y,z, and w that satisfy the following matrix equation:

 

 

By matrix multiplication, this equation can be rewritten as:

 

This matrix equation produces the following system of linear equations all of which were obtained by setting corresponding column and row spaces equal to each other:

 

(1)  2x + z = 2x + y      (2)  2y + w = x     (3)  x = 2z + w     (4)  y = z

 

Equation (4) tells us that y = z so we can substitute y in for z in equations 1,2,and 3. 

 

2x + z = 2x + z     (2)  2z + w = x     (3)  x = 2z + w     (4) y = z

 

Next, we see that equations 2 and 3 are identical equations.  Also note that equation 1 will hold true no matter what values we choose for x and y.  Thus, we are only left with 2 equations that we must take into consideration when choosing our values for matrix B.

 

x = 2z + w and y = z

 

For simplification, let’s let y = a and w = b.  Then y = z = a and x = 2a + b.

 

Placing our values obtained for x,y,z, and w into matrix B the following matrix is produced:

 

Matrix B =   

 

Keep in mind that if a and b are any real numbers, A*B = B*A in this example.  Thus, simply select a real number to substitute in for a, select a real number to substitute in for b, and the matrix B that is produced by your substitution will allow A*B = B*A   In other words, there is an infinite amount of matrices for B that allow A*B = B*A, but all of the matrices for B that allow this to be true must be of the form above.

 

Just for fun, let’s find an example of a matrix B that is of the form above and see if A*B = B*A. 

 

If we let a = 1 and b = 2 (one of the many examples) the following matrix is produced using the form of matrix B above:

 

 

Setting up the equation A * B = B * A yields:

 

 

Sure enough by matrix multiplication we get the following:  (A * B does indeed equal B * A):

 

 

Under each course heading are links to solutions to application problems that I have created.    

 

Elementary Matrix and Linear Algebra

 

This course was taken by me during the summer of 2002, as an independent study, at UW-Madison and was made possible by Dr. Amir Assadi.

 

Links to solutions to application problems that I have created:

 

 

Abstract Algebra

 

This course was taken by me in the Spring and Fall of 2002 at UW-Whitewater and was taught by Dr. Geetha Samaranyake.

 

Links to solutions to application problems that I have created:

 

    

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