Bimodal Diagrams

I think of these as "flippers" - they flip from one state to another. The idea is to present two states of some situation or object as a single diagram with two modes, rather than as two separate "snapshots" of those states which must then be visually integrated.

a) Prism identity

This is a simple matrix multiplication identity used to relate a linear combination of columns to a linear system of equations, and occurs in multiple proofs and derivations (e.g. finding the matrix of a linear transformation, finding the change-of-basis matrix, deriving the dependency algorithm, etc.). The coloured blocks represent columns of numbers and the dots represent single numbers; the colours of both are coordinated to show the structure of the identity. Actual numbers are deliberately not included to avoid cluttering the diagram; what's important is the structure of the identity.

b) Existence of function inverses

These flippers illustrate the situations under which a function does or does not have an inverse.

c) Matrix Operators

This is a sample of a large collection of flippers illustrating the effects of two- and three- dimensional matrix operators on vectors and points (i.e. before and after). Note the use of "ghosts"; ghosts are useful when you want to compare visually the initial and final positions/states of a moving or changing object.