Inverse Transformations

Linear Algebra (Matrices) · Axiom Academy

When can a linear transformation be reversed — and what does its inverse look like? A linear transformation is invertible if there is a second transformation T^ -1 that exactly reverses it: whatever T moves, T^ -1 moves back. Watch shear the unit square into a parallelogram, then watch A^ -1 run every point back to where it began. Apply, then undo — the round trip returns every vector — applying T then T^ -1 changes nothing, and — applying them in the other order changes nothing either. T is invertible exactly when it is bijective — both one-to-one (injective) and onto (surjective). For that translates into four checks on A that all rise or fall together. Watch each one light up for . Domain and codomain have the same dimension — A is . Only a square matrix can have a two-sided inverse. A nonzero determinant means A doesn't collapse area (or volume) to zero — the map is non-degenerate. Only the zero vector maps to . Nothing is crushed together, so T is one-to-one. : the image fills the whole codomain, so T is onto. 3. When the Map Can't Be Undone Now take the singular matrix . Its second column is twice the first, so . Watch it crush the entire plane down onto a single line . Whole families of different inputs land on the same output — once that happens, there is no way to tell them apart, so no inverse can exist.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.