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The main reason why matrix multiplication is defined in a somewhat tricky way is to make matrices represent linear transformations in a natural way.

The main reason why matrix multiplication is defined in a somewhat tricky way is to make matrices represent linear transformations in a natural way.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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The main reason why matrix multiplication is defined in a somewhat tricky way is to make matrices represent linear transformations in a natural way. Let's give an example of a simple linear transformation. Suppose my linear transformation is T(x, y) = (x + y, 2y – x). Imagine (x, y) as a coordinate in 2D space, as usual. This transformation T transforms the point (x, y) to the point (x + y, 2y – a). So, for example. T(-2,1) = (-1,4), T(5, 3) = (8, 1), etc. Now suppose I want a matrix that represents my transformation T. Let's do this by writing the coefficients of x and y as the entries of this matrix. Like this: 1 T = 1 (; )- Now comes the big step: I want to be able to write T(x, y) = (x+y, 2y – x) like this: 7(;) -(,"") x + y 2y – x 1 Since we chose T to be -1 ,we have: GOO-) 1 x + y -a + 2y The left hand side looks suspiciously like the product of two matrices, which should equal the right hand side. For this product to make sense, I need: (* ')()-(**)- x + y and (1 )()-(--»). -x + 2y I think I don't need to take too long to convince you that the only way for this to be possible is to define matrix multiplication as it is usually defined. Bottom line: matrix multiplication is defined the way it is to be able to represent linear transformations in a natural way.
Transcribed Image Text:The main reason why matrix multiplication is defined in a somewhat tricky way is to make matrices represent linear transformations in a natural way. Let's give an example of a simple linear transformation. Suppose my linear transformation is T(x, y) = (x + y, 2y – x). Imagine (x, y) as a coordinate in 2D space, as usual. This transformation T transforms the point (x, y) to the point (x + y, 2y – a). So, for example. T(-2,1) = (-1,4), T(5, 3) = (8, 1), etc. Now suppose I want a matrix that represents my transformation T. Let's do this by writing the coefficients of x and y as the entries of this matrix. Like this: 1 T = 1 (; )- Now comes the big step: I want to be able to write T(x, y) = (x+y, 2y – x) like this: 7(;) -(,"") x + y 2y – x 1 Since we chose T to be -1 ,we have: GOO-) 1 x + y -a + 2y The left hand side looks suspiciously like the product of two matrices, which should equal the right hand side. For this product to make sense, I need: (* ')()-(**)- x + y and (1 )()-(--»). -x + 2y I think I don't need to take too long to convince you that the only way for this to be possible is to define matrix multiplication as it is usually defined. Bottom line: matrix multiplication is defined the way it is to be able to represent linear transformations in a natural way.
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