change of basis
Change of basis is the art of describing the very same vectors and maps using a different set of reference directions. The arrows in space do not move at all; only the numbers we use to name them change, like reading the same place off two different street grids.
Why bother? Because the right coordinate system can make a hard problem easy. A transformation that looks tangled in the standard basis may turn out, in a cleverly chosen basis, to be a simple stretch along each axis. Picking good axes is half the battle in much of applied math.
When you switch coordinate systems, a transformation's matrix takes the form B^(-1) A B, where B holds the new basis vectors as its columns. That sandwich of multiplications translates into the new language, does the work, and translates back. The matrix changes its look, but the underlying map is untouched.
B translates from new coordinates to standard, A acts, and B^(-1) translates back.
Two matrices linked by B^(-1) A B are called similar. Similar matrices are the same transformation seen through different coordinate glasses; they share key traits like determinant and eigenvalues.