Quick question about multiplying matrices:

[Vay]

Let a matrix of a system of linear equations be W and a matrix that it is multiplied by as P. The order of coefficients corresponding to the variables for matrix W would be X, Y, and Z distributed across the row as X for the first column, Y for the second column, and Z for the third; however, this is ordered differently for the matrix of P, where the value of X, Y, and Z are distributed across in rows; where X is for the first row, Y is for the second, and Z is for the third. Is there a reason for the difference in order, or that there is really no reason to it and it is done for the sake of order?

 

In short, why are multiplications of matrices ordered so that the dimensions of the multiplied matrices are that of m*n is multiplied with n*k. Or visually, the variables of matrix M, distributed horizontally, corresponds to the vertical distribution of the same variables of the matrix P, which matrix M is multiplied with.

 

My textbook says this as a rule and doesn't explain why this rule is a rule:

 

"Let A be an m X n matrix and let B be an n X k matrix. The product matrix AB is the m X k matrix whose entry in the ith row and jth column is the product of the ith row of A and the jth column of B."

 

I have taken a guess at the reason, and the reason being that they are completely two different systems of linear equations (systems of linear equations A and B), so that the distributions of coefficient and variables are alternated, where the only similarity is that the horizontal coefficient of system A corresponds to the vertical variables of system B (relative to matrix A). The alteration is then only for the sake of organization?

Edited November 17, 2011 by Vay

[DrRocket]

Let a matrix of a system of linear equations be W and a matrix that it is multiplied by as P. The order of coefficients corresponding to the variables for matrix W would be X, Y, and Z distributed across the row as X for the first column, Y for the second column, and Z for the third; however, this is ordered differently for the matrix of P, where the value of X, Y, and Z are distributed across in rows; where X is for the first row, Y is for the second, and Z is for the third. Is there a reason for the difference in order, or that there is really no reason to it and it is done for the sake of order?

 

In short, why are multiplications of matrices ordered so that the dimensions of the multiplied matrices are that of m*n is multiplied with n*k. Or visually, the variables of matrix M, distributed horizontally, corresponds to the vertical distribution of the same variables of the matrix P, which matrix M is multiplied with.

 

My textbook says this as a rule and doesn't explain why this rule is a rule:

 

"Let A be an m X n matrix and let B be an n X k matrix. The product matrix AB is the m X k matrix whose entry in the ith row and jth column is the product of the ith row of A and the jth column of B."

 

I have taken a guess at the reason, and the reason being that they are completely two different systems of linear equations (systems of linear equations A and B), so that the distributions of coefficient and variables are alternated, where the only similarity is that the horizontal coefficient of system A corresponds to the vertical variables of system B (relative to matrix A). The alteration is then only for the sake of organization?

 

The "rule" is the result of the correspondence between matrix multiplication and linear transformations expressed in terms of a selected basis. This then carries over naturally to the composition of two linear transformations.

 

Your book should show you this in a section on linear transformations. It is a fairly gory exercise in the bookkeeping of indices.

[Vay]

Thanks, but unfortunately my book is on applied mathematics. I will be transferring next year to another school for a major in physics. So hopefully things clear up then when I take calculus courses.

Edited November 17, 2011 by Vay

[DrRocket]

Thanks, but unfortunately my book is on applied mathematics. I will be transferring next year to another school for a major in physics. So hopefully things clear up then when I take calculus courses.

 

You will not find this topic discussed in most calculus courses. You will more usually encounter it in a linear algebra class. Any good text with " Linear Algebra" in the title should have what you need. Linear algebra is commonly encountered after introductory calculus, but calculus is not really a prerequisite.

 

If your text is an applied mathematics text, then the topic may be presented under "matrix theory" and in that context the rule for matrix multiplication is simply a definition.

[imatfaal]

You can pick up copies of Gilbert Strang's Linear Algebra text from ebay and amazon - you can also download (from apple or from mit's distance learning site) videos of the entire course at MIT that it follows.