# Matrices, Real 2×2 Linear Algebra, Labeled Triangles, and Coordinate Transformations: Seven-Round Workflow Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. [← Return to results](/en/paths/) This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Quadratic equations over the reals](/en/mko/mko-quadratic-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quadratic-equation) [Concept guides](/en/derivations/?q=mko-quadratic-equation) - Stage 1 · required ## [Triangle](/en/mko/mko-triangle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-triangle) [Concept guides](/en/derivations/?q=mko-triangle) - Stage 1 · required ## [Euclidean distance in an equal-unit orthonormal frame](/en/mko/mko-distance-formula/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-distance-formula) [Concept guides](/en/derivations/?q=mko-distance-formula) - Stage 1 · required ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 2 · optional ## [Finite matrices: dimensions, indexing, and transpose](/en/mko/mko-matrix/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances. [Related questions](/en/explore/?mko=mko-matrix) [Concept guides](/en/derivations/?q=mko-matrix) - Stage 3 · optional ## [Matrix-represented linear maps on the real plane](/en/mko/mko-linear-transformation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear. [Related questions](/en/explore/?mko=mko-linear-transformation) [Concept guides](/en/derivations/?q=mko-linear-transformation) - Stage 4 · optional ## [Determinants of real 2×2 matrices](/en/mko/mko-determinant/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices. [Related questions](/en/explore/?mko=mko-determinant) [Concept guides](/en/derivations/?q=mko-determinant) - Stage 5 · optional ## [Real eigenvalues and nonzero eigenvectors of 2×2 matrices](/en/mko/mko-eigenvalue/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues. [Related questions](/en/explore/?mko=mko-eigenvalue) [Concept guides](/en/derivations/?q=mko-eigenvalue) - Stage 6 · optional ## [Congruence of nondegenerate triangles with explicit correspondence](/en/mko/mko-congruence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection. [Related questions](/en/explore/?mko=mko-congruence) [Concept guides](/en/derivations/?q=mko-congruence) - Stage 7 · optional ## [Labeled triangle similarity with a specified positive scale](/en/mko/mko-similarity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale. [Related questions](/en/explore/?mko=mko-similarity) [Concept guides](/en/derivations/?q=mko-similarity) - Stage 8 · optional ## [Plane-coordinate isometries and uniform scaling](/en/mko/mko-geometric-transformations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261007%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients. [Related questions](/en/explore/?mko=mko-geometric-transformations) [Concept guides](/en/derivations/?q=mko-geometric-transformations) Path source & statuspath-pulse-foundations-20261007 · candidate [Original paths JSON](/data/architecture/learning-paths.json)