# Wave5 Fourth Layer: Rank-Nullity, Uniform and Absolute Convergence, Estimators, CRT, Spanning Trees and Lp Limits

Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations.

This is a suggested learning sequence, not a universal logical dependency or a formal proof.


- Stage 1 · required



## [Kernel of a linear map](/en/mko/mko-linear-map-kernel/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-linear-map-kernel) [Concept guides](/en/derivations/?q=mko-linear-map-kernel)


- Stage 1 · required



## [Image of a linear map](/en/mko/mko-linear-map-image/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-linear-map-image) [Concept guides](/en/derivations/?q=mko-linear-map-image)


- Stage 1 · required



## [Matrix rank](/en/mko/mko-matrix-rank/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-matrix-rank) [Concept guides](/en/derivations/?q=mko-matrix-rank)


- Stage 1 · required



## [Dimension of a finite-dimensional vector space](/en/mko/mko-vector-space-dimension/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-vector-space-dimension) [Concept guides](/en/derivations/?q=mko-vector-space-dimension)


- Stage 1 · required



## [Pointwise convergence of functions](/en/mko/mko-pointwise-convergence-of-functions/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-pointwise-convergence-of-functions) [Concept guides](/en/derivations/?q=mko-pointwise-convergence-of-functions)


- Stage 1 · required



## [Metric space](/en/mko/mko-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-metric-space) [Concept guides](/en/derivations/?q=mko-metric-space)


- Stage 1 · required



## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value)


- Stage 1 · required



## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier)


- Stage 1 · required



## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-infinite-series) [Concept guides](/en/derivations/?q=mko-infinite-series)


- Stage 1 · required



## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-sequence-limit) [Concept guides](/en/derivations/?q=mko-sequence-limit)


- Stage 1 · required



## [Parametric statistical model](/en/mko/mko-parametric-statistical-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-parametric-statistical-model) [Concept guides](/en/derivations/?q=mko-parametric-statistical-model)


- Stage 1 · required



## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable)


- Stage 1 · required



## [Sample space](/en/mko/mko-sample-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-sample-space) [Concept guides](/en/derivations/?q=mko-sample-space)


- Stage 1 · required



## [Integer](/en/mko/mko-integer/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-integer) [Concept guides](/en/derivations/?q=mko-integer)


- Stage 1 · required



## [Natural-number congruence modulo a positive modulus](/en/mko/mko-modular-congruence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-modular-congruence) [Concept guides](/en/derivations/?q=mko-modular-congruence)


- Stage 1 · required



## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-gcd-coprime) [Concept guides](/en/derivations/?q=mko-gcd-coprime)


- Stage 1 · required



## [Nonempty finite trees: connectivity, cycles, and unique paths](/en/mko/mko-tree/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-tree) [Concept guides](/en/derivations/?q=mko-tree)


- Stage 1 · required



## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph)


- Stage 1 · required



## [Connected component](/en/mko/mko-connected-component/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-connected-component) [Concept guides](/en/derivations/?q=mko-connected-component)


- Stage 1 · required



## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-expectation) [Concept guides](/en/derivations/?q=mko-expectation)


- Stage 1 · required



## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

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



## [Sigma-algebras and measure spaces](/en/mko/mko-sigma-algebra-measure-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-sigma-algebra-measure-space) [Concept guides](/en/derivations/?q=mko-sigma-algebra-measure-space)


- Stage 2 · optional



## [Rank-nullity theorem](/en/mko/mko-rank-nullity-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T).

[Related questions](/en/explore/?mko=mko-rank-nullity-theorem) [Concept guides](/en/derivations/?q=mko-rank-nullity-theorem)


- Stage 3 · optional



## [Uniform convergence of functions](/en/mko/mko-uniform-convergence-of-functions/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously.

[Related questions](/en/explore/?mko=mko-uniform-convergence-of-functions) [Concept guides](/en/derivations/?q=mko-uniform-convergence-of-functions)


- Stage 4 · optional



## [Absolute convergence of series](/en/mko/mko-absolute-convergence-of-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general.

[Related questions](/en/explore/?mko=mko-absolute-convergence-of-series) [Concept guides](/en/derivations/?q=mko-absolute-convergence-of-series)


- Stage 5 · optional



## [Statistical estimator](/en/mko/mko-statistical-estimator/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate.

[Related questions](/en/explore/?mko=mko-statistical-estimator) [Concept guides](/en/derivations/?q=mko-statistical-estimator)


- Stage 6 · optional



## [Chinese remainder theorem](/en/mko/mko-chinese-remainder-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product.

[Related questions](/en/explore/?mko=mko-chinese-remainder-theorem) [Concept guides](/en/derivations/?q=mko-chinese-remainder-theorem)


- Stage 7 · optional



## [Spanning tree](/en/mko/mko-spanning-tree/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph.

[Related questions](/en/explore/?mko=mko-spanning-tree) [Concept guides](/en/derivations/?q=mko-spanning-tree)


- Stage 8 · optional



## [Lp convergence](/en/mko/mko-lp-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261104%2F)

On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero.

[Related questions](/en/explore/?mko=mko-lp-convergence) [Concept guides](/en/derivations/?q=mko-lp-convergence)



Path source & statuspath-pulse-foundations-20261104 · candidate

[Original paths JSON](/data/architecture/learning-paths.json)