Matrix MultiplicationDate: 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/ |
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