# Wave5 Second Layer: Dimension, Kernels, Images, Topological Boundary, Compactness, Bayes and Regular Languages

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



## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Linear independence and bases](/en/mko/mko-linear-independence-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Cardinality comparison of sets](/en/mko/mko-cardinality-comparison/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Linear map between real vector spaces](/en/mko/mko-general-linear-map/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Real vector subspace](/en/mko/mko-vector-subspace/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

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



## [Topological closure and interior](/en/mko/mko-closure-and-interior/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Open cover and subcover](/en/mko/mko-open-cover-and-subcover/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

[Related questions](/en/explore/?mko=mko-open-cover-and-subcover) [Concept guides](/en/derivations/?q=mko-open-cover-and-subcover)


- Stage 1 · required



## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%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



## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Event](/en/mko/mko-event/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Deterministic finite automaton](/en/mko/mko-finite-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 1 · required



## [Decision problem](/en/mko/mko-decision-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

Existing canonical prerequisite for the new objects in this pulse.

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


- Stage 2 · optional



## [Dimension of a finite-dimensional vector space](/en/mko/mko-vector-space-dimension/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality.

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


- Stage 3 · optional



## [Kernel of a linear map](/en/mko/mko-linear-map-kernel/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source.

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


- Stage 4 · optional



## [Image of a linear map](/en/mko/mko-linear-map-image/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span.

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


- Stage 5 · optional



## [Topological boundary](/en/mko/mko-topological-boundary/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement.

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


- Stage 6 · optional



## [Compact topological space](/en/mko/mko-compact-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

A space is compact when every open cover has a finite subcover; every finite topological space is compact.

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


- Stage 7 · optional



## [Bayes theorem](/en/mko/mko-bayes-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods.

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


- Stage 8 · optional



## [Regular language](/en/mko/mko-regular-language/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261102%2F)

A language is regular when a complete DFA recognizes all and exactly its words.

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



Path source & statuspath-pulse-foundations-20261102 · candidate

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