# Seven Rounds of Linear and Finite-Mapping Depth Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Linear equation in one variable](/en/mko/mko-linear-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-linear-equation) [Concept guides](/en/derivations/?q=mko-linear-equation) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 2 · optional ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Use integer vectors in a fixed standard basis: two weighted terms, opposite directional differences, two scalar-addition routes, and candidate component residuals. Zero scalars/vectors are legal; component arithmetic is not an inner product or length. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 2 · optional ## [Finite matrices: dimensions, indexing, and transpose](/en/mko/mko-matrix/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Use complete nonempty 1–5-row/column integer matrices: transpose round trips, both transpose-of-sum routes, three 1-based index guards, and candidate-transpose residuals. Reject mismatched shapes before subtraction; transpose is neither inversion nor element reversal. [Related questions](/en/explore/?mko=mko-matrix) [Concept guides](/en/derivations/?q=mko-matrix) - Stage 2 · optional ## [Injectivity and distinct inputs](/en/mko/mko-injective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Validate complete finite total tables before enumerating every distinct-input collision, deleting an input with original labels retained, guarding appended outputs, or restricting to an explicit source subset. Empty sources are injective; a valid append may create nine labels and collisions still use the original numbering. Representation repetition is not a multivalued function. [Related questions](/en/explore/?mko=mko-injective) [Concept guides](/en/derivations/?q=mko-injective) - Stage 2 · optional ## [Surjectivity and codomain coverage](/en/mko/mko-surjective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Enumerate all declared codomain fibers/missing targets; a changed codomain must contain every existing output before evaluating its surjectivity. Preserve original source labels on deletion and distinguish typed empty fibers from outside-codomain candidates. Empty-to-empty is surjective, empty-to-nonempty is not; actual range is not automatically the declared codomain. [Related questions](/en/explore/?mko=mko-surjective) [Concept guides](/en/derivations/?q=mko-surjective) - Stage 3 · optional ## [Matrix-represented linear maps on the real plane](/en/mko/mko-linear-transformation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Declare the full R²→R² column-vector matrix model Mv: reconstruct basis-image columns, compare additivity/scalar routes, and test the zero-point obstruction for Mv+b. The zero matrix is linear; finite values do not prove arbitrary functions linear. [Related questions](/en/explore/?mko=mko-linear-transformation) [Concept guides](/en/derivations/?q=mko-linear-transformation) - Stage 3 · optional ## [Determinants of real 2×2 matrices](/en/mko/mko-determinant/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Retain signed real 2×2 determinants: swapped-row sign, t² matrix scaling, ordinary row-by-column products in both orders, and absolute-value candidate audits. Nonzero real determinants permit real-field invertibility, not a transfer to integer-ring invertibility; matching product determinants do not make the products equal. [Related questions](/en/explore/?mko=mko-determinant) [Concept guides](/en/derivations/?q=mko-determinant) - Stage 3 · optional ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261016%2F) Use integer-coefficient two-variable/two-equation systems over R: substitute rational candidates into both rows, distinguish solution classes with coefficient/augmented ranks, add a multiple of one complete row to the other, and scale the first row. A zero determinant does not decide inconsistency; zero scaling is not invertible even if a redundant system’s solution set happens to remain unchanged. [Related questions](/en/explore/?mko=mko-system-linear-equations) [Concept guides](/en/derivations/?q=mko-system-linear-equations) Path source & statuspath-pulse-foundations-20261016 · candidate [Original paths JSON](/data/architecture/learning-paths.json)