MATH.18:1 - Problem frame

Use this pattern when two mathematical descriptions appear to address the same construction or question but use different objects, primitive operations, relations or notions of equality. You need to know which calculations, arguments or changes can be carried from one description to the other.

Begin with one consequence you want to transfer. It may be a constructed object, an equation, a solution, or a way of composing operations. An interpretation of that consequence, with its supporting construction or a reason it fails, is the first useful result. A broader equivalence claim requires correspondingly broader comparison.

A mathematical account here consists of the objects, operations, relations and assumptions used to describe the question. An interpretation specifies how to read those ingredients through another account and establishes the consequences needed for the proposed use. The main route requires functions, elementary algebra and reasoning with “for every” and “there exists”. It explains the order and algebra used in the first worked case. The vector-and-matrix case is an optional branch requiring elementary linear algebra.

If a supplied bijection and its transported operations already settle the change of representation, use MATH.7. Use the present method when primitives must be reconstructed, only part of an account transfers, or the round trip needs a structural comparison rather than literal equality.