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VECTOR SPACES

 
Acknowledgements
 
Introduction
      Vector Space Categories
      The Construction of a Vector Space
 
Creation of Vector Spaces and Arithmetic with Vectors
      Construction of a Vector Space
      Construction of a Vector Space with Inner Product Matrix
      Construction of a Vector
      Deconstruction of a Vector
      Arithmetic with Vectors
      Indexing Vectors and Matrices
 
Subspaces, Quotient Spaces and Homomorphisms
      Construction of Subspaces
      Construction of Quotient Vector Spaces
 
Changing the Coefficient Field
 
Basic Operations
      Accessing Vector Space Invariants
      Membership and Equality
      Operations on Subspaces
 
Reducing Vectors Relative to a Subspace
 
Bases
 
Operations with Linear Transformations







DETAILS

 
Introduction

      Vector Space Categories

      The Construction of a Vector Space

 
Creation of Vector Spaces and Arithmetic with Vectors

      Construction of a Vector Space
            VectorSpace(K, n) : Fld, RngIntElt -> ModTupFld
            KModule(K, n) : Fld, RngIntElt -> ModFld
            KMatrixSpace(K, m, n) : Fld, RngIntElt, RngIntElt -> ModMatFld
            Hom(V, W) : ModTupFld, ModTupFld -> ModMatFld
            Example ModFld_CreateQ6 (H28E1)
            Example ModFld_CreateK35 (H28E2)

      Construction of a Vector Space with Inner Product Matrix
            VectorSpace(K, n, F) : Fld, RngIntElt, Mtrx -> ModTupFld

      Construction of a Vector
            elt<V | L> : ModTupFld, List -> ModTupFldElt
            V ! Q : ModTupFld, [RngElt] -> ModTupFldElt
            CharacteristicVector(V, S) : ModTupFld, { RngElt } -> ModTupFldElt
            V ! 0 : ModTupFld, RngIntElt -> ModTupFldElt
            Random(V) : ModTupFld -> ModTupFldElt
            Example ModFld_Vectors (H28E3)
            Example ModFld_Matrices (H28E4)

      Deconstruction of a Vector
            ElementToSequence(u) : ModTupFldElt -> [RngElt]

      Arithmetic with Vectors
            u + v : ModTupFldElt, ModTupFldElt -> ModTupFldElt
            - u : ModTupFldElt -> ModTupFldElt
            u - v : ModTupFldElt, ModTupFldElt -> ModTupFldElt
            x * u : FldElt, ModTupFldElt -> ModTupFldElt
            u / x : ModTupFldElt, FldElt -> ModTupFldElt
            NumberOfColumns(u) : ModTupFldElt -> RngIntElt
            Depth(u) : ModTupRngElt -> RngIntElt
            (u, v) : ModTupFldElt, ModTupFldElt -> FldElt
            IsZero(u) : ModElt -> BoolElt
            Norm(u) : ModTupFldElt -> FldElt
            Normalise(u) : ModTupFldElt -> ModTupFldElt
            Rotate(u, k) : ModTupFldElt, RngIntElt -> ModTupFldElt
            Rotate(~u, k) : ModTupFldElt, RngIntElt ->
            NumberOfRows(u) : ModTupFldElt -> RngIntElt
            Support(u) : ModTupFldElt -> { RngElt }
            TensorProduct(u, v) : ModTupFldElt, ModTupFldElt -> FldElt
            Trace(u, F) : ModTupFldElt, Fld -> ModTupFldElt
            Weight(u) : ModTupFldElt -> RngIntElt
            Example ModFld_Arithmetic (H28E5)
            Example ModFld_InnerProduct (H28E6)

      Indexing Vectors and Matrices
            u[i] : ModTupFldElt, RngIntElt -> RngElt
            u[i] : = x : ModTupFldElt, RngIntElt, RngElt -> ModTupFldElt
            Example ModFld_Indexing (H28E7)

 
Subspaces, Quotient Spaces and Homomorphisms

      Construction of Subspaces
            sub<V | L> : ModTupFld, List -> ModTupFld
            Morphism(U, V) : ModTupFld, ModTupFld -> RModMatElt
            Example ModFld_Subspace1 (H28E8)
            Example ModFld_Subspace2 (H28E9)

      Construction of Quotient Vector Spaces
            quo<V | L> : ModTupFld, List -> ModTupFld, Map
            V / U : ModTupFld, ModTupFld -> ModTupFld, Map
            Example ModFld_Quotients1 (H28E10)
            Example ModFld_Quotients2 (H28E11)
            Example ModFld_Quotients3 (H28E12)

 
Changing the Coefficient Field
      ExtendField(V, L) : ModTupFld, Fld -> ModTupFld, MapHom
      RestrictField(V, L) : ModTupFld, Fld -> ModTupFld, MapHom
      VectorSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

 
Basic Operations

      Accessing Vector Space Invariants
            V . i : ModTupFld, RngIntElt -> ModTupFldElt
            CoefficientField(V) : ModTupFld -> Fld
            Degree(V) : ModTupFld -> RngIntElt
            Degree(u) : ModTupFldElt -> RngIntElt
            Dimension(V) : ModTupFld -> RngIntElt
            Generators(V) : ModTupFld -> { ModElt }
            NumberOfGenerators(M) : ModTupFld -> RngIntElt
            OverDimension(V) : ModTupFld -> RngIntElt
            OverDimension(u) : ModTupFldElt -> RngIntElt
            Generic(V) : ModFld -> ModFld
            Parent(V) : ModFld -> SetPow

      Membership and Equality
            v in V : ModTupFldElt, ModTupFld -> BoolElt
            v notin V : ModTupFldElt, ModTupFld -> BoolElt
            U subset V : ModTupFld, ModTupFld -> BoolElt
            U notsubset V : ModTupFld, ModTupFld -> BoolElt
            U eq V : ModTupFld, ModTupFld -> BoolElt
            U ne V : ModTupFld, ModTupFld -> BoolElt

      Operations on Subspaces
            U + V : ModTupFld, ModTupFld -> ModTupFld
            U meet V : ModTupFld, ModTupFld -> ModTupFld
            U meet:= V : ModTupFld, ModTupFld -> ModTupFld
            &meet S : [ ModTupFld ] -> ModTupFld
            TensorProduct(U, V) : ModTupFld, ModTupFld -> FldElt
            Complement(V, U) : ModTupFld, ModTupFld -> ModTupFld
            Transversal(V, U): ModTupFld, ModTupFld -> { ModTupFldELt }

 
Reducing Vectors Relative to a Subspace
      ReduceVector(W, v) : ModTupRng, ModTupRngElt -> ModTupRngElt
      ReduceVector(W, ~v) : ModTupRng, ModTupRngElt ->
      DecomposeVector(U, v) : ModTupRng, ModTupRngElt -> ModTupRngElt, ModTupRngElt

 
Bases
      VectorSpaceWithBasis(Q) : [ModTupFldElt] -> ModTupFld
      Basis(V) : ModTupFld -> [ModTupFldElt]
      BasisElement(V, i) : ModTupFld, RngIntElt -> ModTupFldElt
      BasisMatrix(V) : ModTupFld -> ModMatElt
      Coordinates(V, v) : ModTupFld, ModTupFldElt -> [FldElt]
      Dimension(V) : ModTupFld -> RngIntElt
      ExtendBasis(Q, U) : [ModTupFldElt], ModTupFld -> [ModTupFldElt]
      ExtendBasis(U, V) : ModTupFld, ModTupFld -> [ModTupFldElt]
      IsIndependent(S) : { ModTupFldElt } -> BoolElt
      IsIndependent(Q) : [ ModTupFldElt ] -> BoolElt
      Example ModFld_Basis (H28E13)

 
Operations with Linear Transformations
      v * a : ModTupFldElt, ModMatFldElt -> ModTupFldElt
      a * b : ModMatRngElt, ModMatRngElt -> ModMatRngElt
      Domain(a) : ModMatRngElt -> ModTupRng
      Codomain(a) : ModMatRngElt -> ModTupRng
      Image(a) : ModMatRngElt -> ModTupRng, Map, Map
      Rank(a) : ModMatRngElt -> RngIntElt
      Kernel(a) : ModMatRngElt -> ModTupFld, Map
      Cokernel(a) : ModMatRngElt -> ModTupFld, Map
      Example ModFld_LinearTrans (H28E14)

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013