Free Modules#
- Introduction
- Definition of a Module
- Accessing Module Information
M . i: ModTupRng, RngIntElt → ModEltCoefficientRing(M): ModTupRng → RngBaseRing(M): ModTupRng → RngCoefficientRing(M): ModRng → RngBaseRing(M): ModRng → RngCoefficientField(M): ModFld → FldBaseField(M): ModFld → FldGenerators(M): ModTupRng → { ModTupRngElt }OverDimension(M): ModTupRng → RngIntEltOverDimension(u): ModTupRngElt → RngIntEltModuli(M): ModTupRng → [ RngElt ]Parent(u): ModTupRngElt → ModRngGeneric(M): ModRng → ModRng
- Standard Constructions
- Construction of Elements
elt< M | a₁, ..., aₙ >: ModTupRng, List → ModTupRngEltM ! Q: ModTupRng, [RngElt] → ModTupRngEltCharacteristicVector(M, S): ModRng, { RngIntElt } → ModRngEltZero(M): ModRng → ModRngEltM ! 0: ModRng, RngIntElt → ModRngEltRandom(M): ModRng → ModRngEltExample: Elements- Deconstruction of Elements
- Operations on Module Elements
- Properties of Vectors
- Inner Products
- Bases
- Submodules
- Construction of Submodules
- Operations on Submodules
- Membership and Equality
u in M: ModTupRngElt, ModTupRng → BoolEltu in M: ModMatRngElt, ModMatRng → BoolEltu notin M: ModTupRngElt, ModTupRng → BoolEltu notin M: ModMatRngElt, ModMatRng → BoolEltN subset M: ModTupRng, ModTupRng → BoolEltN subset M: ModMatRng, ModMatRng → BoolEltN notsubset M: ModTupRng, ModTupRng → BoolEltN notsubset M: ModMatRng, ModMatRng → BoolEltN eq M: ModTupRng, ModTupRng → BoolEltM eq N: ModMatRng, ModMatRng → BoolEltN ne M: ModTupRng, ModTupRng → BoolEltM ne N: ModMatRng, ModMatRng → BoolElt
- Operations on Submodules
- Quotient Modules
- Homomorphisms
- \({\operatorname{Hom}}_{R}(M, N)\) for \(R\)-modules
- \({\operatorname{Hom}}_{R}(M, N)\) for Matrix Modules
- Modules \({\operatorname{Hom}}_{R}(M, N)\) with Given Basis
- The Endomorphism Ring
- The Reduced Form of a Matrix Module
- Construction of a Matrix
- Element Operations
u * a: ModTupRngElt, ModMatRngElt → ModTupRngElta(u): ModTupRngElt, ModMatRngElt → ModTupRngElta * b: ModMatRngElt, ModMatRngElt → ModMatRngElta ^ -1: ModMatRngElt, RngIntElt → ModMatRngEltCodomain(S): ModMatRng → ModTupRngCodomain(a): ModMatRngElt → ModTupRngCokernel(a): ModMatRngElt → ModTupRngDomain(S): ModMatRng → ModTupRngDomain(a): ModMatRngElt → ModTupRngImage(a): ModMatRngElt → ModTupRngKernel(a): ModMatRngElt → ModTupRngNullSpace(a): ModMatRngElt → ModTupRngMorphism(M, N): ModTupRng, ModTupRng → ModMatRngEltRank(a): ModMatRngElt → RngIntEltIsBijective(a): ModMatRngElt → BoolEltIsInjective(a): ModMatRngElt → BoolEltIsSurjective(a): ModMatRngElt → BoolEltExample: Operations