OCME · Bilingual mathematics & derivations
Matrices, Real 2×2 Linear Algebra, Labeled Triangles, and Coordinate Transformations: Seven-Round Workflow
On this page
- Arithmetic operations
- Real number
- Plane vectors in a fixed basis
- Function mapping
- Quadratic equations over the reals
- Triangle
- Euclidean distance in an equal-unit orthonormal frame
- Ratio and proportion
- Natural-number powers and principal square roots
- Coordinate plane
- Finite matrices: dimensions, indexing, and transpose
- Matrix-represented linear maps on the real plane
- Determinants of real 2×2 matrices
- Real eigenvalues and nonzero eigenvectors of 2×2 matrices
- Congruence of nondegenerate triangles with explicit correspondence
- Labeled triangle similarity with a specified positive scale
- Plane-coordinate isometries and uniform scaling
Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Plane vectors in a fixed basis
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Quadratic equations over the reals
Existing canonical prerequisite for the new objects in this pulse.
Triangle
Existing canonical prerequisite for the new objects in this pulse.
Euclidean distance in an equal-unit orthonormal frame
Existing canonical prerequisite for the new objects in this pulse.
Ratio and proportion
Existing canonical prerequisite for the new objects in this pulse.
Natural-number powers and principal square roots
Existing canonical prerequisite for the new objects in this pulse.
Coordinate plane
Existing canonical prerequisite for the new objects in this pulse.
Finite matrices: dimensions, indexing, and transpose
Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances.
Matrix-represented linear maps on the real plane
Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear.
Determinants of real 2×2 matrices
This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices.
Real eigenvalues and nonzero eigenvectors of 2×2 matrices
An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues.
Congruence of nondegenerate triangles with explicit correspondence
In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection.
Labeled triangle similarity with a specified positive scale
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.
Plane-coordinate isometries and uniform scaling
Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients.
Path source & status
path-pulse-foundations-20261007 · candidate