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alexander
@@ -132,6 +132,26 @@
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)
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]
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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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// Vektorräume
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@@ -270,6 +290,19 @@
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]
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// Rekursive Folgen
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Rekursive Folgen]
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E.g: $a_1 x_(n-1) + a_2 x_(n) = x_(n+1)$
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1. Als Matrix Schreiben $F: vec(x_(n-1), x_(n)) = vec(x_n, x_(n+1))$ \
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$F s_(n-1) = s_(n)$
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2. Diagonaliseren: $F = R D R^(-1) $ \
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3. Wiederholte Anwendung: $F^n = R D^n R^(-1)$
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]
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#colbreak()
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#bgBlock(fill: colorAbbildungen)[
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@@ -345,7 +378,7 @@
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$kern(Phi) := {v in V | Phi(v) = 0}$
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- $A ve(x) = ve(0)$ (Lösung des Homogenen Gleichungssystem)
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*Dimensionssatz:* Sei $A$ lineare Abbildung \
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*Dimensionssatz:* lineare Abbildung $A: V -> W$ \
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$dim(V) = dim(kern(A)) + dim(Bild(A))$ \
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$dim(V) = dim(kern(A)) + Rang(A)$
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@@ -388,16 +421,12 @@
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Nur bei $A in RR^(n times n)$ möglich
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Bei qudratischen $A:$ \ $n equiv "#Spalten" equiv dim ve(x)$
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$Rang(A) = n$
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],
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image("../images/linAlg/Gauss1.jpg", width: 2cm),
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[
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*Nullzeile*:
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Pro Nullzeile eine frei Var $t, s, ...$
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Bei qudratischen $A:$ \ $Rang(A) = n - "#Nullzeilen"$
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],
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image("../images/linAlg/Gauss3.jpg", width: 2cm),
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@@ -405,6 +434,9 @@
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*Wiederspruch*: Keine Lösung
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]
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)
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$Rang(A) = "#Nicht-Null-Zeilen"$ \
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$=$ Anzahl der linear unabhänigen Zeilen/Spalten
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$kern(A) = dim ve(x) - Rang(A)$ (Dimensionssatz)
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@@ -624,32 +656,31 @@
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- Winkel: $cos alpha = ip(v, w)/(norm(v) norm(w))$
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*Orthogonal Vektoren:* $ip(v, w) = 0$ \
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*Ortho#text(red)[normal] Vektoren:* $ip(v, w) = 0$ UND $norm(v),norm(w) = 1$
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*Ortho#text(red)[normal] Vektoren:* \
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$ip(v, w) = 0 "UND" norm(v),norm(w) = 1, p(v, v) = ip(w, w) = 1$
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#SeperatorLine
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*Orthogonal Projektion* $pi_U(v) = limits(sum)_(i=1)^k ip(v, u_i) u_i$
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$U subset V$ Untervektorraum eines Eukldische VRs $V$, \ $U$ in orthogo#text(red)[normal] Basis:
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$U subset V$ Untervektorraum eines Eukldische VRs $V$, \ $U$ in orthogo#text(red)[normal] Basis
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Orthogonals Kompliment: $kern(pi_U) = U^tack.t$
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]
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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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#bgBlock(fill: colorVR)[
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#subHeading(fill: colorVR)[Unitäre Vektorräume]
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alles $in CC$
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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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*Sequilinearform:* $ip(x, y)$
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- $ip(x, y)$ linear im #underline("1.") Argument
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- $ip(v, w_1 + w_2) = ip(v, w_1) + ip(v, w_2) = \
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ip(v, lambda w) = overline(lambda) ip(v, w)$
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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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*Hermitische Form:* $ip(v, w) = ip(w, v)$
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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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*Hermitische Skalarprodukt:* $in RR$
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]
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@@ -792,20 +823,10 @@
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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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#sinTable
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Rekursive Folgen]
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E.g: $a_1 x_(n-1) + a_2 x_(n) = x_(n+1)$
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1. Als Matrix Schreiben $F: vec(x_(n-1), x_(n)) = vec(x_n, x_(n+1))$ \
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$F s_(n-1) = s_(n)$
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2. Diagonaliseren: $F = R D R^(-1) $ \
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3. Wiederholte Anwendung: $F^n = R D^n R^(-1)$
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]
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// Lineare Differenzialgleichungen
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#bgBlock(fill: colorMatrix)[
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#subHeading(fill: colorMatrix)[Lineare Differenzialgleichungen]
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@@ -839,29 +860,27 @@
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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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*Invertierbar wenn:*
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- $A$ bijektiv
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- $A x = b$ eindeutig
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- $A x = 0$ nur $ve(0)$
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- Spalten/Zeilen 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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- $0$ ist keine Eigenwert
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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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- Immer Bijektiv
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- $det (A) = plus.minus 1$
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- $O^T = O^(-1)$
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- $O^T O = O O^T = I$
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- $ip(O v, O w) = ip(v, w)$
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],
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[*Unitair* $equiv$ Orthogonal $in CC$], [
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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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