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b/src/cheatsheets/LinearAlgebra.typ @@ -45,6 +45,7 @@ *Halbgruppe:* $(M, compose): M times M arrow M$ - Assoziativgesetz: $a dot (b dot c) = (a dot b) dot c$ \ z.B. Komposition von Funktionen + - Abgeschlossenheit *Monoid* Halbgruppe $M$ mit: - Neutrales-/Identäts-Element: $e in M : a e = e a = a$ *Kommutativ/abelsch:* Halbgruppe/Monoid mit @@ -295,9 +296,12 @@ - Bei Matrix: $Bild(M) = spann("Spalten Vektoren")$ - $op("Bild") Phi := {Phi in W | v in V}$ - *Rang* + *Rang:* $op("Rang") f := dim op("Bild") f$ - Bei Matrizen: \ $Rang(f) <= min(n, m) equiv min("#Spalten", "#Zeilen")$ + - $Rang("Zeilen Vektoren") = Rang("Spalten Rang")$ + - $Rang(A) = Rang(A^T)$ + - $"#Linear unabhäniger Spalten/Zeilen"$ *Nullraum/Kern:* \ $kern(Phi) := {v in V | Phi(v) = 0}$ @@ -315,7 +319,7 @@ - Homogense Lineares Gleichungsystem: $A ve(x) = ve(0) $ Lösungsmenge: $LL = kern(A)$, immer: $ve(0) in L$ \ - - In-Homogense LGS: $A ve(x) = ve(b) $ + - In-Homogense LGS: $A ve(x) = ve(b) $<< #SeperatorLine @@ -336,16 +340,16 @@ ], ) - - #grid(columns: (auto, 1fr), - row-gutter: 1mm, + row-gutter: 3mm, column-gutter: 2mm, image("../images/linAlg/Gauss2.jpg", width: 2cm), [ *Eindeutige Lösung* $-->$ Normale Rückeinsetzung - $n equiv "#Spalten" equiv dim ve(x)$ + Nur bei $A in RR^(n times n)$ möglich + + Bei qudratischen $A:$ \ $n equiv "#Spalten" equiv dim ve(x)$ $Rang(A) = n$ ], @@ -355,6 +359,7 @@ *Nullzeile*: Pro Nullzeile eine frei Var $t, s, ...$ + Bei qudratischen $A:$ \ $Rang(A) = n - "#Nullzeilen"$ ], image("../images/linAlg/Gauss3.jpg", width: 2cm), @@ -362,6 +367,14 @@ *Wiederspruch*: Keine Lösung ] ) + + $kern(A) = dim ve(x) - Rang(A)$ (Dimensionssatz) + + #SeperatorLine + *Matrix Invertieren* + #image("../images/linAlg/InverseMatrix.jpg") + + $KK^(2 times 2): A = mat(a, b; c, d) \ A^(-1) = 1/det(A) mat(d, -b; -c, a) = 1/(a d - b c) mat(d, -b; -c, a) $ ] @@ -375,6 +388,10 @@ - Matrix-Matrix: $A dot B = Phi_A compose Phi_B = Phi_A (Phi_B (ve(x)))$ \ $c_(j i) = sum^n_(s=1) a_(j s) b_(s i)$ + $(KK^(n times n), +)$ sind Gruppe, $(KK^(n times n), dot)$ sind Monoid, + + #image("../images/linAlg/matMul.jpg") + #SeperatorLine #grid(columns: (1fr, 1fr, 1fr), @@ -394,15 +411,14 @@ $(A^T)^T = A$, $(A dot B)^T = B^T dot A^T$ ) + ] - - ] + #colbreak() // Matrix Typem #bgBlock(fill: colorMatrix)[ #let colred(x) = text(fill: red, $#x$) #let colblue(x) = text(fill: blue, $#x$) - #subHeading(fill: colorMatrix)[Matrix Typen] #align(center, scale($colred(m "Zeilen") colblue(n "Splate")\ A in KK^(colred(m) times colblue(n))$, 120%)) #grid(columns: (1fr, 1fr), $quad mat( @@ -449,6 +465,7 @@ *Invertierbar wenn:* \ $A$ bijektiv, $det(A) = 0$ \ $"Spalten Vekoren lin. unabhänig"$ \ + $Rang(A) = n, A in KK^(n times n)$ \ $det(A) = 0$ \ *Nicht Invertierbar wenn:*\ @@ -479,6 +496,118 @@ #bgBlock(fill: colorAbbildungen)[ #subHeading(fill: colorAbbildungen)[Determinate und Bilinearform] + + - Sclar-Produkt $ip(ve(a), ve(b))$ ist Bi-Linearform + - Symetrisch + - Determinante einer Matrix: $det(A in RR^(m times n))$ ist $n$-Linearform (sogar alternierend) + + *$k$-Linearform:* Lineare $f: KK^n times KK^n times ... -> KK$ + - Für $k=2 : $ Bi-Linerform + - Linearität: \ + $f(ve(v)_1, lambda ve(v)_2) = lambda f(ve(v)_1, ve(v)_2) \ + f(ve(v)_1, ve(x) + ve(y)) = f(ve(v)_1, ve(x)) + f(ve(v)_1, ve(y)) + $ + - *Symetrisch* wenn: $f(ve(v)_1, ve(v)_2) = f(ve(v)_2, ve(v)_1), space space forall ve(v)_1, ve(v)_2 in KK^n$ + + - *Alternierend* wenn: $f(ve(v), ve(v)) = 0, space space forall ve(v) in KK^n$ + - $f(ve(v)_1, ... #text(red, $ve(v)_i$), #text(blue, $ve(v)_j$), ... ve(v)_k) = -f(ve(v)_1, ... #text(blue, $ve(v)_j$), #text(red, $ve(v)_i$), ... ve(v)_k) $ + - Tauschung von Argumenten $->$ Vorzeichen Flip + - $ve(v)_1, ... "linear abhänig" <=> f(ve(v)_1, ...) = 0$ + - $ve(v)_1, ... "linear unabhänig" <=> f(ve(v)_1, ...) != 0$, eindeutig + + #SeperatorLine + + *Determinaten Form* \ + Nicht tiviale ($f(...) = 0$) n-Linearform auf einem VR. + $exists$ Immer, in jeder Scalierung + + Speziell für Martizen $in KK^(n times n)$ \ (Qudratische, Endomorphismus) + + *Leibniz-Formel* + + $det(A) = limits(sum)_(sigma in S_n) sign(sigma)( a_(sigma(1)1) dot a_(sigma(2)2) dot dots dot a_(sigma(n)n))$ + + $S_n := "Alle Permutation von n Element" $ \ + $sign(sigma) = (-1)^"#Vertauschungen"$ \ + Anzahl der Vertauschungen, die nötig sind um $sigma$ von $(1, 2, 3, ...)$ zu erzeugen \ + $sigma(n): n$-te Element aus der Tupel $sigma$ + + *Bsp:* $A in KK^(n times n)$ + #grid( + columns: 9, + row-gutter: 1mm, + $S_3 =$, ${$, $(1,2,3),$, $(1,3,2),$, $(2,1,3),$, $(2,3,1),$, $(3,1,2),$, $(3,2,1)$, $}$, + "#Vert.", $$, + align(center, $0$), + align(center, $1$), + align(center, $1$), + align(center, $2$), + align(center, $2$), + align(center, $1$), $$, + $sign(sigma)$, $$, + align(center, $1$), + align(center, $-1$), + align(center, $-1$), + align(center, $1$), + align(center, $1$), + align(center, $-1$), + ) + + $det(A) = &(a_11 a_22 a_33) - (a_11 a_23 a_32) - (a_12 a_21 a_33) + \ + &(a_12 a_23 a_31) + (a_13 a_21 a_32) - (a_13 a_22 a_31) + \ + $ + + #SeperatorLine + + *Regel von Saurus* + #grid( + columns: (auto, 1fr), + image("../images/linAlg/saurus.png", height: 1cm), + align(center+horizon, $= a_11 a_22 a_33 + a_12 a_23 a_31 + a_13 a_21 a_32 \ + - a_13 a_22 a_31 - a_12 a_21 a_33 - a_11 a_23 a_32 + $) + ) + + *Laplace Entwicklung* + + #grid( + columns: (auto, 1fr), + column-gutter: 2mm, + image("../images/linAlg/laplace.jpg", height: 2cm), + [ + - Nach Spalte oder Zeile Entwicklung + - #underline([Auf Vorzeichen Achten!!!]) + - Zeilen mit vielen Nuller wählen + ] + ) + + *Determinate Tricke-Kiste* + + - Orthogonal-Matrix: $det(A) = 1$ + - Diagonal-/Oberdreiecks-/Unterdreick-Matrix: \ $det(A) = product a_(i i)$ + - $det(A) = product lambda_(i i) "(Eigenwerte)"$ + + - Partionen: $mat(A, B; 0, C) slash mat(A, 0; B, C) -> det(A) = det(A) dot det(C)$ + + #grid( + columns: (1fr, 1fr), + column-gutter: 2mm, + row-gutter: 2mm, + $A,B in KK^(n times n) :$, $$, + $det(A dot B) = det A dot det B$, + $det(A^(-1)) = 1/det(A)$, + $det(A) = det(A^T)$, + $det(lambda A) = lambda^n det(A)$, + $det mat(a, b; c, d) = a d - b c$, + ) + + Elementare (Gauß) Zeilen Umfohrungen kann man machen + + NICHT Qudratischen Matrizen $det(D)$ nicht definiert, Nur für Endomorphisen + + $det(A) = 0 <=> "invertierbar" <=> "bijektiv"$ + + ] #bgBlock(fill: colorVR)[ @@ -604,9 +733,9 @@ 1. $A A^T$ berechnen $A A^T in RR^(m times m)$ - 2. $A A^T$ diagonalisieren in $R$, $D$ + 2. $A A^T$ diagonalisieren in $R$, $D$ (dabei EWs, EVs berechen) - 3. Singulärwere berechen: $sigma_i = sqrt(lambda_i) $ + 3. 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