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linear algebra: motivation for matrix multiplication ?
I have been studying linear algebra out of a proof-based text, and it has thus far offered much insight into matrices, such as the fundamentals upon which determinants have been established. The only problem that I have is that it has not given me a reason for WHY matrix multiplication is defined the way that it is. I would like to know the motivation behind defining matrix multiplication in the way that it is defined.
- JBLv 71 decade agoFavorite Answer
It is natural to describe the function F:R^2 ->R^2 that carries (x,y) to (ax+by, cx+dy) by the matrix
[ a b ]
[ c d ]
and the function G:R^2 ->R^2 that carries (x,y) to (ex + fy, gx + hy) by the matrix:
[ e f ]
[ g h ]
What would the matrix for the composition (GoF)R^2 -> R^2, defined by
(GoF)(x,y) = G(F(x,y))? Well, it would be nice to denote it symbolically by
[ e f ] [ a b ]
[ g h] [ c d ] ..... ..... ..... ...(*)
But you can do the algebra and figure out that
G(F(x,y)) = G((ax+by, cx+dy)) = (e(ax+by) + f(cx+dy), g(ax+by) + h(cx+dy)) = ((ea+fc)x + (eb+fd)y, (ga+hc)x + (gb+hd)y)
But the matrix of that is
[ (ea+fc) (eb+fd) ]
[ (ga+hc) (gb+hd) ] ..... ..... ..... ..... (**)
So we better define the product of the matices (*) above as this matrix (**).
In short, matrix multiplication is defined the way it is so that matrix multiplication will work consistently with composition of linear transformations.