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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.

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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.

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