login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

Edit detail for SandBoxComplementsdalgebrelineaire revision 8 of 9

1 2 3 4 5 6 7 8 9
Editor: Bill Page
Time: 2008/02/01 11:27:08 GMT-8
Note: correction to [[ as advised by Martin

changed:
-         [[- 1,0,1,0,0,0],[- -,- -,0,- -,1,0],[- -,- -,0,-,0,1]]
         @[[- 1,0,1,0,0,0],[- -,- -,0,- -,1,0],[- -,- -,0,-,0,1]]

Download: pdf dvi ps src tex log

axiom
--- --- linalg.input --- Francois Maltey - janvier 2008 --- --- --- A partir de rowEchelon --- sameSizeVectors? Lb == null Lb => true n := #(first Lb) every? (t +-> #t=n, rest Lb)
Type: Void
axiom
basis mat == mat2 := rowEchelon mat basis := [] indrow : Integer := 1 n : Integer := ncols mat m : Integer := nrows mat for k in 1..n repeat if indrow <= m and mat2.(indrow,k) ~= 0 then basis := cons (column (mat, k), basis) indrow := indrow + 1 reverse basis
Type: Void
axiom
basisLV Lv == null Lv => [] not (sameSizeVectors? Lv) => error "vectors have not the same size" basis transpose matrix Lv
Type: Void
axiom
basisMat mat == basis mat
Type: Void
axiom
sumBasisLLV LLv == basisLV concat LLv
Type: Void
axiom
sumBasis2 (Lv1, Lv2) == basisLV concat (Lv1, Lv2)
Type: Void
axiom
kernelMat mat == lv := nullSpace mat #lv = 1 and lv.1 = 0*lv.1 => [] lv
Type: Void
axiom
subVector (v, a, b) == vector (elt (entries v, a..b))
Type: Void
axiom
linearVector (t, Lv) == reduce (+, [t.i*Lv.i for i in 1..#t])
Type: Void
axiom
intBasis2 (Lv1, Lv2) == Lb1 := basisLV Lv1 Lb2 := basisLV Lv2 null Lb1 => [] null Lb2 => [] #(first Lb1) ~= #(first Lb2) => error "vectors have not the same size" lkv := kernelMat transpose matrix concat (Lb2, Lb1) d1 := #Lb1 d2 := #Lb2 LcoeffV1 := [subVector (kv, d2+1, d1+d2) for kv in lkv] [linearVector (cc, Lb1) for cc in LcoeffV1]
Type: Void
axiom
intBasisLLV LLv == #LLv = 0 => error "no space to intersect" #LLv = 1 => LLv.1 --reduce (intBasis2, LLv) intBasis2 (LLv.1, intBasisLLV rest LLv)
Type: Void

axiom
--- --- inversegeneralisee.input --- Francois Maltey - janvier 2008 --- --- inverse generalisee --- --- a partir du livre Algebre lineaire par Joseph Grifone p.375 --- applique pas à pas la méthode du livre à l'exemple A := matrix [[2,1,-1,1],[1,1,0,1],[3,2,-1,2]]
LatexWiki Image(1)
Type: Matrix Integer
axiom
--- kerA est une base de ker f ou f est definie par la matrice A KerA := kernelMat A
axiom
Compiling function kernelMat with type Matrix Integer -> List Vector
      Integer
LatexWiki Image(2)
Type: List Vector Integer
axiom
--- Le noyau de la matrice dont les lignes sont des vecteurs generateurs --- d'un sev est le sev orthogonal. C'est un sous-espace supplementaire. --- baseF est une base d'un sous-espace supplémentaire de ker f baseF := kernelMat matrix KerA
LatexWiki Image(3)
Type: List Vector Integer
axiom
--- les vecteurs colonnes de A engendre l'image Im f. --- baseImA est une base de l'image de f calculee a partir de la matrice A baseImA := basis A Function definition for basis is being overwritten. The type of the local variable basis has changed in the computation. We will attempt to interpret the code. Compiled code for basis has been cleared.
LatexWiki Image(4)
Type: List Vector Integer
axiom
--- baseG et baseG2 sont deux bases d'un sous-espace supplementaire de Im f --- l'une calculee a partir de la matrice A, l'autre d'une base de Im f. baseG := kernelMat transpose A
LatexWiki Image(5)
Type: List Vector Integer
axiom
baseG2:= kernelMat matrix baseImA
LatexWiki Image(6)
Type: List Vector Integer
axiom
--- La restriction g de f est un isomorphisme du supplementaire F de ker f --- dans Im f. La matrice de g dans les bases de F et de Im f est obtenue --- en decomposant dans Im f les images des vecteurs de la base de F. --- La commande particularSolution effectue cette decomposition. MP := transpose matrix baseImA
LatexWiki Image(7)
Type: Matrix Integer
axiom
map (X +-> A*X, baseF)
LatexWiki Image(8)
Type: List Vector Integer
axiom
map (X +-> particularSolution (MP, A*X), baseF)
LatexWiki Image(9)
Type: List Union(Vector Fraction Integer,"failed")
axiom
B := transpose matrix map (X +-> particularSolution (MP, A*X), baseF)
LatexWiki Image(10)
Type: Matrix Fraction Integer
axiom
--- La matrice C est celle de l'isomorphisme reciproque dans ces bases. C := B^-1
LatexWiki Image(11)
Type: Matrix Fraction Integer
axiom
--- la projection orthogonale de E' sur Im f peut être obtenue --- à partir d'une base orthonormee de Im f. --- baseImA est une base de Im f, la fonction gramschmidt construit --- la bon associee. GS := gramschmidt (baseImA::List Vector Expression Integer)
LatexWiki Image(12)
Type: List Matrix Expression Integer
axiom
bonImA := map (M +-> column (M, 1), GS)
LatexWiki Image(13)
Type: List Vector Expression Integer
axiom
projortho X == reduce (+, map (V +-> dot (V, X::Vector Expression Integer) * V, bonImA))
Type: Void
axiom
--- exemple de projection orthogonale sur Im f. projortho vector [1,2,4]
axiom
Compiling function projortho with type Vector PositiveInteger -> 
      Vector Expression Integer
LatexWiki Image(14)
Type: Vector Expression Integer
axiom
--- construction de la composition g^-1 o p de E' dans E --- en calculant les images des vecteurs de la base canonique de E' --- puis passage des coordonnees de g^-1 o p de la base de F --- à la base canonique de E. Id3 := diagonalMatrix [1 for i in 1..3]
LatexWiki Image(15)
Type: Matrix Integer
axiom
RES1 := [projortho column (Id3, i) for i in 1..3]
axiom
Compiling function projortho with type Vector Integer -> Vector 
      Expression Integer
LatexWiki Image(16)
Type: List Vector Expression Integer
axiom
RES2 := [X::Vector Fraction Integer for X in RES1]
LatexWiki Image(17)
Type: List Vector Fraction Integer
axiom
RES3 := [particularSolution (MP, X) for X in RES2]
LatexWiki Image(18)
Type: List Union(Vector Fraction Integer,"failed")
axiom
RES := [C * X for X in RES3::List Vector Fraction Integer]
LatexWiki Image(19)
Type: List Vector Fraction Integer
axiom
AA := transpose matrix baseF * transpose matrix RES
LatexWiki Image(20)
Type: Matrix Fraction Integer
axiom
--- vérifications de quelques propriétés : A AA A = A et AA A AA = AA --- transpose (A AA) = A AA : car les sev choisis sont des orthogonaux AA * A * AA - AA
LatexWiki Image(21)
Type: Matrix Fraction Integer
axiom
A * AA * A - A
LatexWiki Image(22)
Type: Matrix Fraction Integer
axiom
transpose (AA * A) - AA * A
LatexWiki Image(23)
Type: Matrix Fraction Integer
axiom
transpose (A * AA) - A * AA
LatexWiki Image(24)
Type: Matrix Fraction Integer