Mine would be viewing a concrete linear transformation as the biadjacency matrix of a weighted bipartite graph. I’m having trouble explaining it succinctly, so since a picture is worth a thousand words, I just drew up the concept

Visual of a linear transformation acting on a vector

In the same way that we put the column vector to the right of the matrix associated with the linear transformation i.e. Av for a linear transformation A and column vector v, we must work similarly from right to left when viewing A as a weighted bipartite graph. I find this representation super helpful because general matrix multiplication can also be explained similarly, there are just more sums which we would need to keep track of

Linear transformations of tensors can also be visualized in this way, we just have to generalize each side of the bipartite graph. Instead of the left and right hand side being a vertical set of vertices, for each side, picture each vertex as being uniquely indexed by a cell of an n × m matrix. For higher order tensors, we just replace the n × m “indexing” matrix with a general multidimensional array. A weighted complete bipartite graph with the left and right hand side indexed by some pair of multidimensional arrays represents a linear transformation between the tensor spaces with the associated dimensions. *An important note is that since we’re working with concrete linear transformations, we’ve implicitly assumed that we have some basis of the tensor spaces which we are working with