From 8c37369dff38cac7382b083f961dd6fc496a14af Mon Sep 17 00:00:00 2001 From: alexander Date: Fri, 20 Feb 2026 20:39:30 +0100 Subject: [PATCH] Added Darstellungsmatrix --- src/cheatsheets/LinearAlgebra.typ | 114 ++++++++++++++++++++--- src/images/linAlg/BasisWechsel.jpg | Bin 0 -> 148103 bytes src/images/linAlg/DarstellungsMatrix.jpg | Bin 0 -> 135564 bytes src/images/linAlg/DarstelsMatrix.jpg | Bin 0 -> 97973 bytes 4 files changed, 103 insertions(+), 11 deletions(-) create mode 100644 src/images/linAlg/BasisWechsel.jpg create mode 100644 src/images/linAlg/DarstellungsMatrix.jpg create mode 100644 src/images/linAlg/DarstelsMatrix.jpg diff --git a/src/cheatsheets/LinearAlgebra.typ b/src/cheatsheets/LinearAlgebra.typ index abbd58e..66188e2 100644 --- a/src/cheatsheets/LinearAlgebra.typ +++ b/src/cheatsheets/LinearAlgebra.typ @@ -143,6 +143,11 @@ $B_1, B_2, ...$ Erzeugerssystem vom gleichen $V$ \ $=> abs(B_1)=abs(B_2)...$ + + Vektor dratstellung durch Basis Vektoren: \ + $ve(v) = lambda_1 ve(b_1) + lambda_2 ve(b_2) + ...$ + - $lambda_1, lambda_2, ...$ beschreiben ein #underline[eindeutig] Punk + *Basisergänzungssatz:* Sei $M = {ve(v_1), ... ve(v_n)}, ve(v_i) in V$ lin. unabhänig aber $M$ kein Basis des $V$. Dann $exists v_(n+1)$ sodass $M union {ve(v_(n+1))}$ lin unabhänig (aber evt. eine Basis ist) #SeperatorLine @@ -168,14 +173,75 @@ - Kodimension: $dim V - dim U$ - Wenn $dim U = dim V <=> U = V "(Kodimension"=0")"$ - Kodimension $= 1$ Hyperebend - #colbreak() ] #bgBlock(fill: colorVR)[ #subHeading([Darstellungs Matrix], fill: colorVR) + + Matrix $equiv$ Linera Abbildung \ + + Sclar-Matrix Multiplikation $lambda M$ \ + - Kommutativ, Assoziativgesetz, (keine Gruppe wege fehlender Abgeschlossenheit) + + Matrix-Matrix Addtion $M + N$ + - Kommutativ Gruppe $(KK^(n times n), +)$ + + Matrix-Matrix Multiplikation/Composition \ + $M dot N equiv Phi_M compose Phi_N = Phi_M (Phi_N (ve(x)))$ \ + $c_(j i) = sum^n_(s=1) a_(j s) b_(s i)$ + + + #SeperatorLine *Vektorraum Isomorphismus* - $V tilde.equiv W <=> dim(V) = dim(W)$ - $V tilde.equiv W <=> exists f: V -> W, f "bijektiv (umkehrbar)"$ + + *Koordinatensystem* Ein bestimmte Wahl von Basisvektoren/Basis $ve(b_1), ve(b_2), ...$ + + #SeperatorLine + #grid( + columns: (auto, 1fr), + column-gutter: 2mm, + image("../images/linAlg/BasisWechsel.jpg", height: 1.3cm), + + [ + Vektorraum $V tilde.equiv KK^n$ (in Basis $A$)\ + Vektorraum $V tilde.equiv KK^n$ (in Basis $B$)\ + + $Phi_A, Phi_B$ Bijektiv Mappings zwischen $V$ und dem $KK^m slash KK^n$ + ] + ) + + $space_A T_B$: Basiswechsel: $K^n$ (in Basis $A$) $->$ $K^n$ (in Basis $B$) + $space_B T_A$: Basiswechsel: $K^n$ (in Basis $B$) $->$ $K^n$ (in Basis $A$) + + Wenn $V, KK^n "(in Basis A/B)"$ ein $RR^n slash CC^n$ \ ist $Phi_(A slash B) = mat(dots.v, dots.v; ve(b_1), ve(b_2), ...; dots.v, dots.v,)$, $ve(b_1), ve(b_2), ...$ \ Basisvektoren der Basis von $A slash B$ + + #SeperatorLine + *Darstellungs-Matrix* + + Idee: Wir führen Abbildung $f$ nicht $A -> B$ sonderem in $KK^n -> KK^m$ durch $-->$ Darstellungs-Matrix $D$ + + #grid( + columns: (auto, 1fr), + column-gutter: 2mm, + image("../images/linAlg/DarstellungsMatrix.jpg", height: 1.6cm), + [ + $f: V -> W$ Orignal Abbildung \ + Vektorraum $V tilde.equiv KK^n$ (in Basis $A$)\ + Vektorraum $V tilde.equiv KK^n$ (in Basis $B$)\ + + $Phi_A, Phi_B$ Bijektiv Mappings zwischen $V$ und dem $KK^m, KK^m$ + ], + ) + + #grid(columns: (1fr, 1fr), + $D = Psi^(-1) compose f compose Phi$, + $$ + ) + + #SeperatorLine + ] #colbreak() @@ -238,20 +304,24 @@ *Bild:* Wertemenge $WW$ - $f(I subset A) = B$ (Oft $I = A$) + - Bei Matrix: $Bild(A) = spann("Spalten Vektoren")$ - Basis $B : op("spann")(B)$ - $op("Bild") Phi := {Phi in W | v in V}$ - *Nullraum/Kern:* \ - $op("Kern") Phi := {v in V | Phi(v) = 0}$ - *Rang* $op("Rang") f := dim op("Bild") f$ + *Nullraum/Kern:* \ + $kern(Phi) := {v in V | Phi(v) = 0}$ + - $A ve(x) = ve(0)$ + *Dimensionssatz:* Sei $A$ lineare Abbildung \ $dim(V) = dim(kern(A)) + dim(Bild(A))$ \ - $dim(V) = dim(kern(A)) + Rang(A)$ \ + $dim(V) = dim(kern(A)) + Rang(A)$ - $dim(V) = dim(Bild(A)) "oder" dim(kern(A)) = 0 \ <=> A "bijektiv" <=> "invertierbar"$ + #linebreak() + + $"Wenn" dim(V) = dim(Bild(A)) "oder" dim(kern(A)) = 0 \ <=> A "bijektiv" <=> "invertierbar"$ ] #bgBlock(fill: colorAbbildungen)[ @@ -430,7 +500,22 @@ #bgBlock(fill: colorMatrixVerfahren)[ #subHeading(fill: colorMatrixVerfahren)[Schur-Zerlegung] - immer anwendbar; + Wenn dsa charakteristische Polynom $chi_A "von" A in CC^(n times n) slash RR^(n times n) "in" chi_A(lambda) = (lambda_1 - lambda)(lambda_2 - lambda)... "zerfällt"$ dann ist Schur-Zerlegung möglich + - Gilt für $CC^(n times n)$ immer + + #grid( + columns: (1fr, 3fr), + $R = B^* A B$, + [ + $B:$ orthogonal/unitair $KK^(n times n)$ \ + $R:$ Oberedreiecks Matrix $KK^(n times n)$ \ + $B^* = B^T "für" RR, B^* = B^(-T) "für" CC$ + ] + ) + + - $A,R$ haben die selben Eigenwerte + - Schur-Zerlegung ist nicht eindeutig, (Diagnoal elemen bis auf die Reihnfolge schon) + - Wenn $A$ diagonaliserbar $=>$ $R$ Dignoalmatrix ] #bgBlock(fill: colorMatrixVerfahren)[ @@ -456,7 +541,8 @@ 5. $S in RR^(n times m)$ (wie $A$): \ $S = mat(sigma_0, 0; 0, sigma_1; dots.v, dots.v; 0, 0) quad quad quad S = mat(sigma_0, 0, dots, 0; 0, sigma_1, ..., 0)$ ] - + + #colbreak() #bgBlock(fill: colorMatrix)[ #subHeading(fill: colorMatrix)[Matrix Normen] @@ -469,8 +555,8 @@ *Frobenius-Norm* $||A||_"M" = sqrt(sum_(i=1)^m sum_(j=1)^n a_(m n)^2)$ - *Induzierte Norm* $||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V)$\ - $ = sup_(||v|| = 1) (||A v||_V)/(||v||_V)$ + *Induzierte Norm* \ + $||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V) = sup_(||v|| = 1) (||A v||_V)/(||v||_V)$ - submultiplikativ - verträglich mit einer Vektornorm $||dot||_V$ @@ -491,7 +577,13 @@ ] #bgBlock(fill: colorMatrix)[ - #subHeading(fill: colorMatrix)[Differenzialgleichungen] + #subHeading(fill: colorMatrix)[Lineare Differenzialgleichungen] + + $y'_1(t) &= alpha_11 &dot y_1(t) + alpha_12 dot y_2(t) + ...\ + y'_1(t) &= alpha_11 &dot y_1(t) + alpha_12 dot y_2(t) + ... \ + &dots.v &dots.v\ + y'_1(t) &= alpha_11 &dot y_1(t) + alpha_12 dot y_2(t) + ... + $ ] #colbreak() diff --git a/src/images/linAlg/BasisWechsel.jpg b/src/images/linAlg/BasisWechsel.jpg new file mode 100644 index 0000000000000000000000000000000000000000..e30944bbe4b069d9f8441993eddc65425b03da7e GIT binary patch literal 148103 zcmeFZ2UOF~mpA-Hs;D#p=|PHsfFPiBA|fE7pdej|G-=X7iXlNzniK&6MT!WBNQv}L z=paRUmy*zXLJg$whTrZ!yXQIYzGwG6`|SU>=R5c0RVT&U%&|pxTWdm_#6NpJOHi&06+&&ggF9KWQp7b;38K5IP>MqzqC+Gexdvq 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