Mathematica has special sparse-array technology for efficiently handling arrays with literally astronomical numbers of elements when only a small fraction of the elements are nonzero. Mathematica's linear algebra and all standard list operations can be done in symbolic form on SparseArray objects, in which nonzero elements are specified by giving rules—potentially including patterns—for the values of elements at particular positions.
SparseArray — create a sparse array of any rank from a list of rules or a dense array
Normal — convert a sparse array to a dense array of explicit lists
ArrayRules — find position→value rules for elements in a sparse array
CoefficientArrays — convert symbolic equations to sparse arrays
ArrayPlot — visualize a 2D sparse array using grayscales or other color maps
MatrixPlot — visualize a sparse matrix with color, ticks, etc.
GraphPlot — take a sparse array as the adjacency matrix for a graph