Moved stuff around
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Build Typst PDFs (Docker) / build-typst (push) Successful in 38s
This commit is contained in:
alexander
@@ -38,7 +38,7 @@
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}
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#columns(5, gutter: 2mm)[
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// Gruppen
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#bgBlock(fill: colorGruppen)[
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#subHeading(fill: colorGruppen)[Gruppen]
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@@ -92,11 +92,14 @@
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$(S_n, compose)$ ist Gruppe, aber nicht kommutativ
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]
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/*
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// Komplex Zahlen
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Komplexe Zahlen]
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#ComplexNumbersSection()
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]
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*/
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// Matrix Typen
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#bgBlock(fill: colorMatrix)[
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@@ -129,6 +132,8 @@
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)
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]
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#colbreak()
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// Vektorräume
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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Vektorräume (VR)]
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@@ -264,40 +269,6 @@
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Normalweiße alle Abbildung/Matrizen in Kannoischer Basis $hat(e)_1 = vec(1, 0, dots.v), hat(e)_2 = vec(0, 1, dots.v), ...$
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]
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// Matrix Basics
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Matrix Basics]
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Linera Abbildung $equiv$ EINER eindeutige Matrix \
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- Sclar/Matrix: $lambda dot A$
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- Matrix/Matrix: $A + B$
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- Matrix-Matrix: $A dot B = Phi_A compose Phi_B = Phi_A (Phi_B (ve(x)))$ \
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$c_(j i) = sum^n_(s=1) a_(j s) b_(s i)$
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$(KK^(n times n), +)$ sind Gruppe, $(KK^(n times n), dot)$ sind Monoid,
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#image("../images/linAlg/matMul.jpg")
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#SeperatorLine
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#grid(columns: (1fr, 1fr, 1fr),
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row-gutter: 2mm,
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align(center, $(lambda mu) A = lambda (mu A)$),
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grid.cell(colspan: 2, align(center, $(lambda + mu) A = lambda A + mu A$)),
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align(center, $$),
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grid.cell(colspan: 2, align(center, $lambda (A + B) = lambda A + lambda B$)),
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)
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*Transponieren*
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#grid(columns: (1fr, 1fr),
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row-gutter: 2mm,
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$(A + B)^T = A^T + B^T$,
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$(lambda A)^T = lambda A^T$,
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$(A^T)^T = A$,
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$(A dot B)^T = B^T dot A^T$
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)
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]
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#colbreak()
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@@ -444,9 +415,44 @@
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$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) $
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]
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// Matrix Basics
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Matrix Basics]
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Linera Abbildung $equiv$ EINER eindeutige Matrix \
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#colbreak()
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- Sclar/Matrix: $lambda dot A$
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- Matrix/Matrix: $A + B$
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- Matrix-Matrix: $A dot B = Phi_A compose Phi_B = Phi_A (Phi_B (ve(x)))$ \
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$c_(j i) = sum^n_(s=1) a_(j s) b_(s i)$
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$(KK^(n times n), +)$ sind Gruppe, $(KK^(n times n), dot)$ sind Monoid,
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#image("../images/linAlg/matMul.jpg")
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#SeperatorLine
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#grid(columns: (1fr, 1fr, 1fr),
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row-gutter: 2mm,
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align(center, $(lambda mu) A = lambda (mu A)$),
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grid.cell(colspan: 2, align(center, $(lambda + mu) A = lambda A + mu A$)),
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align(center, $$),
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grid.cell(colspan: 2, align(center, $lambda (A + B) = lambda A + lambda B$)),
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)
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*Transponieren*
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#grid(columns: (1fr, 1fr),
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row-gutter: 2mm,
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$(A + B)^T = A^T + B^T$,
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$(lambda A)^T = lambda A^T$,
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$(A^T)^T = A$,
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$(A dot B)^T = B^T dot A^T$
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)
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]
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]
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#columns(5, gutter: 2mm)[
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#bgBlock(fill: colorAbbildungen)[
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#subHeading(fill: colorAbbildungen)[Linearform]
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@@ -523,7 +529,7 @@
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*Regel von Saurus*
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#grid(
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columns: (auto, 1fr),
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image("../images/linAlg/saurus.png", height: 1cm),
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image("../images/linAlg/saurus.png", height: 0.6cm),
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align(center+horizon, $= a_11 a_22 a_33 + a_12 a_23 a_31 + a_13 a_21 a_32 \
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- a_13 a_22 a_31 - a_12 a_21 a_33 - a_11 a_23 a_32
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$)
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@@ -534,7 +540,7 @@
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#grid(
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columns: (auto, 1fr),
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column-gutter: 2mm,
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image("../images/linAlg/laplace.jpg", height: 2cm),
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image("../images/linAlg/laplace.jpg", height: 1.6cm),
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[
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- Nach Spalte oder Zeile Entwicklung
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- #underline([Auf Vorzeichen Achten!!!])
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@@ -587,10 +593,17 @@
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#SeperatorLine
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*Norm*
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- $norm(ve(v)) = 0 <=> ve(v) = ve(0)$
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- $norm(lambda ve(v)) = abs(lambda) norm(ve(v))$
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- Dreieckesungleichung: $norm(x + y) <= norm(x) + norm(y)$
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#grid(
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columns: (auto, auto),
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column-gutter: 3mm,
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[*Norm*],
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[
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- $norm(ve(v)) = 0 <=> ve(v) = ve(0)$
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- $norm(lambda ve(v)) = abs(lambda) norm(ve(v))$
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- Dreieckesungleichung: $norm(x + y) <= norm(x) + norm(y)$
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]
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)
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*Generisch/$L_p$-Norm*: $|| v ||_p = root(p, sum_(k=1)^n (x_k)^p)$
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@@ -620,8 +633,29 @@
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]
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#colbreak()
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// Matrix Normen
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Matrix Normen]
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$|| dot ||_M$ Matrix Norm, $|| dot ||_V$ Vektornorm
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- submultiplikativ: $||A B||_"M" <= ||A||||B||$
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- verträglich mit einer Vektornorm: $||A v||_"V" <= ||A||_"M" ||v||_"V"$
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*Frobenius-Norm* $||A||_"M" = sqrt(sum_(i=1)^m sum_(j=1)^n a_(m n)^2)$
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*Induzierte Norm* \
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$||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V) = sup_(||v|| = 1) ||A v||_V$
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- submultiplikativ
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- verträglich mit einer Vektornorm $||dot||_V$
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*maximale Spaltensumme* $||A||_r = max_(1<= i <= n) sum_(j=1)^n |a_(j)|$
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]
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#colbreak()
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// Eigenwert und Eigenvektoren
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Eigenwert und Eigenvektoren ]
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@@ -664,8 +698,9 @@
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]
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// Gram-Schmit ONB
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Gram-Schmit ONB]
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#subHeading(fill: colorMatrixVerfahren)[Gram-Schmit]
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Idee: $ip("Orth"#text(red)[normal] ve(v), ve(x)) = "Anteil von" ve(x) "an" ve(v)$ \
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Ziel: Basis $W -->$ Orthogonal Basis $V$
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@@ -679,6 +714,7 @@
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]
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// Diagonalisierung
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#bgBlock(fill: colorMatrixVerfahren)[
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#subHeading(fill: colorMatrixVerfahren)[Diagonalisierung]
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$A = R D R^(-1)$
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@@ -756,23 +792,6 @@
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$S = mat(sigma_0, 0; 0, sigma_1; dots.v, dots.v; 0, 0) quad quad quad S = mat(sigma_0, 0, dots, 0; 0, sigma_1, ..., 0)$
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]
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Matrix Normen]
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$|| dot ||_M$ Matrix Norm, $|| dot ||_V$ Vektornorm
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- submultiplikativ: $||A B||_"M" <= ||A||||B||$
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- verträglich mit einer Vektornorm: $||A v||_"V" <= ||A||_"M" ||v||_"V"$
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*Frobenius-Norm* $||A||_"M" = sqrt(sum_(i=1)^m sum_(j=1)^n a_(m n)^2)$
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*Induzierte Norm* \
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$||A||_"M" = sup_(v in V without {0}) (||A v||_V)/(||v||_V) = sup_(||v|| = 1) ||A v||_V$
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- submultiplikativ
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- verträglich mit einer Vektornorm $||dot||_V$
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*maximale Spaltensumme* $||A||_r = max_(1<= i <= n) sum_(j=1)^n |a_(j)|$
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]
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Rekursive Folgen]
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@@ -799,69 +818,65 @@
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#colbreak()
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// Table
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#table(
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columns: (auto, 1fr),
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inset: 2mm,
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fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
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[*Einheits Matrix*\ $I,E$], [
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$det(E) = 1$
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],
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[*Diagonalmatrix* \ $Sigma,D$], [
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Nur Einträger auf Hauptdiagonalen \
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$det(D) = d_00 dot d_11 dot d_22 dot ...$
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],
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[*Symetrisch*\ $S$], [
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$S = S^T$, $S in KK^(n times n)$\
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$A A^T$, $A^T A$ ist symetrisch \
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$S$ immer diagonaliserbar \
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EW immer $in RR$, EV orthogonal
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],
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[*Invertierbar*], [
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$exists A^(-1) : A A^(-1) = A^(-1) A = E$ \
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#sinTable
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]
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#columns(2)[
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#table(
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columns: (auto, 1fr),
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inset: 2mm,
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fill: (x, y) => if (calc.rem(y, 2) == 0) { tableFillLow } else { tableFillHigh },
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[*Einheits Matrix*\ $I,E$], [
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$det(E) = 1$
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],
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[*Diagonalmatrix* \ $Sigma,D$], [
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Nur Einträger auf Hauptdiagonalen \
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$det(D) = d_00 dot d_11 dot d_22 dot ...$
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],
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[*Symetrisch*\ $S$], [
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$S = S^T$, $S in KK^(n times n)$\
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$A A^T$, $A^T A$ ist symetrisch \
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$S$ immer diagonaliserbar \
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EW immer $in RR$, EV orthogonal
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],
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[*Invertierbar*], [
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$exists A^(-1) : A A^(-1) = A^(-1) A = E$ \
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*Invertierbar wenn:* \
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$A$ bijektiv, $det(A) = 0$ \
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$"Spalten Vekoren lin. unabhänig"$ \
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$Rang(A) = n, A in KK^(n times n)$ \
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$det(A) = 0$ \
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*Nicht Invertierbar wenn:*\
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$exists$ EW $!= 0 => not "invertierbar"$
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Keine Qudratische Matrix
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],
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[*Orthogonal*], [
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#grid(
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columns: (1fr, 1fr),
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[
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- Immer Bijektiv
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- $det (A) = plus.minus 1$
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$O^T = O^(-1) quad quad O^T O = O O^T = I$ \
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$ip(O v, O w) = ip(v, w)$
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]
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)
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],
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[*Unitair* $equiv$ Orthogonal $in CC$], [
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- Immer Bijektiv
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$V = V^*$
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],
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[*Diagonaliserbar*], [
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$exists A = B D B^(-1)$, $D$ diagonal,
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$B$: Splaten sind EV von $A$
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- Selbst-Adujunkte ($$) diagonalisierbar
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- Symetrisch Matrix
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- $A in KK^(n times n) "UND" alg(lambda) = geo(lambda)$
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],
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[*postiv-semi-definit*], [
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$forall$ EW $>= 0$
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],
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)
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*Invertierbar wenn:* \
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$A$ bijektiv, $det(A) = 0$ \
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$"Spalten Vekoren lin. unabhänig"$ \
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$Rang(A) = n, A in KK^(n times n)$ \
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$det(A) = 0$ \
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*Nicht Invertierbar wenn:*\
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$exists$ EW $!= 0 => not "invertierbar"$
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Keine Qudratische Matrix
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],
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[*Orthogonal*], [
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#grid(
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columns: (1fr, 1fr),
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[
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- Immer Bijektiv
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- $det (A) = plus.minus 1$
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$O^T = O^(-1) quad quad O^T O = O O^T = I$ \
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$ip(O v, O w) = ip(v, w)$
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]
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)
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],
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[*Unitair* $equiv$ Orthogonal $in CC$], [
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- Immer Bijektiv
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$V = V^*$
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],
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[*Diagonaliserbar*], [
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$exists A = B D B^(-1)$, $D$ diagonal,
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$B$: Splaten sind EV von $A$
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- Selbst-Adujunkte ($$) diagonalisierbar
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- Symetrisch Matrix
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- $A in KK^(n times n) "UND" alg(lambda) = geo(lambda)$
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],
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[*postiv-semi-definit*], [
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$forall$ EW $>= 0$
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],
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)
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]
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Reference in New Issue
Block a user