OCME · Bilingual mathematics & derivations
Wave5 Second Layer: Dimension, Kernels, Images, Topological Boundary, Compactness, Bayes and Regular Languages
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- Real vector space
- Linear independence and bases
- Cardinality comparison of sets
- Linear map between real vector spaces
- Real vector subspace
- Equation
- Function mapping
- Topological closure and interior
- Set operations
- Topological space
- Open cover and subcover
- Quantifiers and explicit domains
- Conditional probability in finite models
- Event
- Probability on finite sample spaces
- Deterministic finite automaton
- Set
- Sequence
- Decision problem
- Dimension of a finite-dimensional vector space
- Kernel of a linear map
- Image of a linear map
- Topological boundary
- Compact topological space
- Bayes theorem
- Regular language
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.
Real vector space
Existing canonical prerequisite for the new objects in this pulse.
Linear independence and bases
Existing canonical prerequisite for the new objects in this pulse.
Cardinality comparison of sets
Existing canonical prerequisite for the new objects in this pulse.
Linear map between real vector spaces
Existing canonical prerequisite for the new objects in this pulse.
Real vector subspace
Existing canonical prerequisite for the new objects in this pulse.
Equation
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Topological closure and interior
Existing canonical prerequisite for the new objects in this pulse.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Topological space
Existing canonical prerequisite for the new objects in this pulse.
Open cover and subcover
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Conditional probability in finite models
Existing canonical prerequisite for the new objects in this pulse.
Event
Existing canonical prerequisite for the new objects in this pulse.
Probability on finite sample spaces
Existing canonical prerequisite for the new objects in this pulse.
Deterministic finite automaton
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Decision problem
Existing canonical prerequisite for the new objects in this pulse.
Dimension of a finite-dimensional vector space
The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality.
Kernel of a linear map
The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source.
Image of a linear map
The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span.
Topological boundary
The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement.
Compact topological space
A space is compact when every open cover has a finite subcover; every finite topological space is compact.
Bayes theorem
With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods.
Regular language
A language is regular when a complete DFA recognizes all and exactly its words.
Path source & status
path-pulse-foundations-20261102 · candidate