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
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