Re viewing matrices as graphs Under the usual matrices-as-linear-maps interpretation, a nxm matrix is a linear map from the free vector space on m elements to

Re viewing matrices as graphs Under the usual matrices-as-linear-maps interpretation, a nxm matrix is a linear map from the free vector space on m elements to the free vector space on n elements, M : Vect(Y) -> Vect(X), where X and Y are as in the article. For a regular map of finite sets Y -> X, we can draw a picture showing how each element of Y goes to an element of X by drawing a bunch of dots for Y and a bunch of dots for X with an arrow from each dot in Y to its image in X. This is a directed bipartite graph where each dot in Y is the source of exactly one edge. How is a linear map, Vect(Y) -> Vect(X), different? Each element of Y goes not to a single element of X, but to a linear combination of elements of X. If y goes to a x1 + b x2, we can think of this as two arrows out of y, each one now being weighted, one arrow to x1 with weight a and one to x2 with weight b (considering edges with weight 0 to be non-existent if you like). This is the graph shown in the article. This interpretation shows why matrix multiplication is path composition. It's exactly analogous to how for the map of finite sets, you find the image of z by first following its arrow to y in Y, then following its arrow to x in X. Here you first follow z to all its images in Y, then you follow those to all their images in X. Relation composition is only a special case. And of course the connection between XxY->R and Vect(Y)->Vect(X) is the usual correspondence between a rank (1,1) tensor and a linear map. Re probabilities: this corresponds to imagining a random process that transforms elements of Y into elements of X where y turns into x with probability p(x,y). The probability that you end up with x (the marginal probability) is the probability that any dot in Y transformed into x, ie. sum over all the arrows that wind up at x.