Matrix Multiplication

Date: 12/18/98 at 19:56:09
From: James Brossard
Subject: Vector Multiplication

Hi,

I am currently reading a linear algebra book, but I am not precisely 
sure of the method of matrix multiplication. That is, why must a row 
vector be multiplied by a column vector? Is the purpose of this 
multiplication just to plug various linear combinations into the A 
matrix? Someone gave me an explanation that the row vector is space 
and a column vector is dual space; however, I have been stuck on this 
question for some time, so I can't visualize dual space or n-space. 
Are these just different dimensions? 

Thanks,
James

Date: 12/19/98 at 07:17:58
From: Doctor Jerry
Subject: Re: Vector Multiplication

Hi James,

Matrix multiplication can be explained or motivated in several ways. 
One of the most basic is through transformations of coordinates.

Suppose that (x,y) are the coordinates of a point in a plane, relative 
to a given coordinate system. Suppose that new coordinates (x',y') are 
assigned (rotation would be one new coordinate system; elastic 
deformation might be another) and they are related to the old by:

   x' = a11*x + a12*y
   y' = a21*x + a22*y

Suppose the new system is transformed again, so that:

   x'' = b11*x' + b12*y'
   y'' = b21*x' + b22*y'

One might seek to figure out coords (x'',y'') for a given (x,y), a 
combination of these two, leading to:

   x'' = b11*x' + b12*y' = b11*(a11*x+a12*y) + b12*(a21*x+a22*y)
   y'' = b21*x' + b22*y' = b21*(a11*x+a12*y) + b22*(a21*x+a22*y)

If this is rearranged, one finds the "formula" for matrix 
multiplication.

One writes, for example:

  [ x' ]   [ a11    a12 ] [ x ]
  [    ] = [            ] [   ]
  [ y' ]   [ a21    a22 ] [ y ]

or this can be written as:

   X' = A*X
   X'' = B*X'

and then:

   X'' = B*(A*X) = (B*A)*X

The agreements about column vs. row come by an attempt to simplify 
these transformations. This "basic" explanation has nothing much to do 
with dual spaces. 

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/