Bestimme den Vektor $v=\left( \begin{array}{} 4 \\ -3 \\ 2 \end{array} \right)$ als Linearkombination der folgenden Menge $M$:

\begin{align*}M=\left\{\left(\begin{array}{l}
1 \\
1 \\
1
\end{array}\right),\left(\begin{array}{l}
1 \\
1 \\
0
\end{array}\right),\left(\begin{array}{l}
1 \\
0 \\
0
\end{array}\right)\right\}\end{align*}

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viewBox="0 86.3 875 382"><use xlink:href="#MJX-8005-TEX-S4-239F" transform="scale(1, 0.924)"></use></svg></g></g></g></g></svg></mjx-container> als Linearkombination der folgenden Menge <mjx-container class="MathJax" jax="SVG"><svg alt="M" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="2.378ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1051 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-8006-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-8006-TEX-I-1D440"></use></g></g></g></svg></mjx-container>:

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1 \\
1 \\
1
\end{array}\right),\left(\begin{array}{l}
1 \\
1 \\
0
\end{array}\right),\left(\begin{array}{l}
1 \\
0 \\
0
\end{array}\right)\right\}\end{align*}" style="vertical-align: -3.733ex" xmlns="http://www.w3.org/2000/svg" width="26.701ex" height="8.597ex" role="img" focusable="false" viewBox="0 -2150 11801.9 3800" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-8007-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 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Prüfe ob der Vektor $v=\left( \begin{array}{c} 0 \\ -\sqrt{2} \\ \sqrt{2} \end{array} \right)$ ein Element des folgenden Vektorraums ist:

\begin{align*}V=\left\{x \in \mathbb{R}^{3} \ \bigg| \ x=\lambda_{1}\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right)+\lambda_{2}\left(\begin{array}{l}
1 \\
1 \\
0
\end{array}\right) ; \lambda_{1}, \lambda_{2} \in \mathbb{R}\right\}\end{align*}

Gegeben sind die Vektoren:

\begin{align*}\begin{array}{l}
a_{1}=\left(\begin{array}{l}
1 \\
0 \\
1
\end{array}\right), a_{2}=\left(\begin{array}{l}
1 \\
1 \\
0
\end{array}\right), a_{3}=\left(\begin{array}{l}
0 \\
1 \\
1
\end{array}\right) \\ \\
b_{1}=\left(\begin{array}{l}
1 \\
0 \\
0
\end{array}\right), b_{2}=\left(\begin{array}{l}
1 \\
2 \\
0
\end{array}\right), b_{3}=\left(\begin{array}{l}
1 \\
2 \\
3
\end{array}\right)
\end{array}\end{align*}

Stellen Sie den Vektor $c=a_{1}+a_{2}+a_{3}$ als Linearkombination der Vektoren $b_{1}, b_{2}, b_{3}$ dar.

Ist der Vektor $x=\left(\begin{array}{c} -3 \\ 4 \\ 7\end{array}\right)$ in der linearen Hülle $\langle u, v, w \rangle$ erhalten? Falls ja bestimme eine Linearkombination. 

\begin{align*}\langle u, v, w \rangle =\left\langle\left(\begin{array}{c}
1 \\
2 \\
1
\end{array}\right),\left(\begin{array}{c}
-2 \\
1 \\
3
\end{array}\right), \left(\begin{array}{c}
4 \\
3 \\
-1
\end{array}\right)\right \rangle\end{align*}

Es seien die Vektoren $a_{1}, a_{2}, a_{3}, a_{4} \in \mathbb{R}^{4}$ mit
\begin{align*}
a_{1}=\left(\begin{array}{c}
1 \\1 \\-1 \\-2
\end{array}\right), \quad a_{2}=\left(\begin{array}{l}
2 \\1 \\0 \\3
\end{array}\right), \quad a_{3}=\left(\begin{array}{c}
0 \\-1 \\2 \\7
\end{array}\right), \quad a_{4}=\left(\begin{array}{c}
-1 \\2 \\-1 \\1
\end{array}\right)
\end{align*}
gegeben. Prüfe ob die folgenden Vektoren $u, v$ als Linearkombinationen von $a_{1}, a_{2}, a_{3}, a_{4}$ darstellbar sind und berechne ggf, diese Darstellung.
\begin{align*}
u=\left(\begin{array}{c}
6 \\-4 \\2 \\-2
\end{array}\right), \quad v=\left(\begin{array}{c}
1 \\3 \\-1 \\2
\end{array}\right)
\end{align*}

Linearkombination, Lineare Hüllen

Eindimensionale Analysis

Quicksort

Serlo 

Metrische Räume

Sortieralgorithmen

Reibung (Gleitreibung)

(Weg)zusammenhängende Räume

Wahr/Falsch

Fundamentalgruppen

Algorithmen

Mannigfaltigkeiten

Informatik

Satz von Green

(Lokal) Kompakte Räume

Satz von Stokes

Satz von Gauß

Beweisaufgabe

reelle Zahlen

Kompakt oder nicht?

Mehrfachintegrale berechnen

Supremum und Infimum

Vektorbündel

Uneigentliches Supremum und Infimum 

Produkt- und Quotientenräume

Vektoranalysis und Mehrfachintegrale

Elektrotechnik

LR-Zerlegung

Gleichstrom

Folgen

Lyapunov-Stabilität

Konvergenz und Divergenz

Beobachtbarkeit

(Matrix-) Normen berechnen

Grundlagen

Grundlagen der Wellenausbreitung

Ladung, Strom, Stromdichte, Spannung

Häufungspunkte und Cauchy-Folgen

Steuerbarkeit

Reihen

Nyquist-Kriterium

Kriterien für Konvergenz von Reihen 

MIMO-System

Stetigkeit von Funktionen

Elektrische Feldgrößen 

Thermodynamik und Kinetik chemischer Reaktionen

Ableitung

Sensitivität

Integrale

Sprungfunktion

Linearität einer Abbildung prüfen

Wärmetransport – Temperaturstrahlung

Implizite Funktionen

Kirchhoffschen Gesetze

Energieerhaltung für geschlossene Systeme

Ideales Gas

Bilanzen offener Systeme

Wärmeenergie 

Strahlung und Lichtempfindung

Reihenschaltung, Parallelschaltung

Überarbeitet 

Thermodynamik und Wärme (Kurzweil)

Reelle Zahlen 

Kinetische Theorie

Schwerpunkte paralleler angreifender Kräfte

Schwingung

Quanten und Materiewellen

Laplace-Transformation

Resultierende Kräfte und Momente ermitteln

Zustandsänderungen 

Grundlagen: Energieniveaus und Feinstruktur

Schwingungen

Grundlagen der Molekülspektroskopie

Elastostatik, Festigkeitslehre

Kernspin und NMR-Spektroskopie

Grundarten der Belastung

Resultierende und Zerlegung von Kräften

Zug und Druck

Stetigkeit (mehrdimensional)

Geometrische Optik

Funktionen (mehrdimensionale)

Kinetik, Kinematik

Kraft- und Drehmomente

Bohr’sches Atommodell

Wärmeübertragung

Haftung (Haftreibung)

Rotation um eine feste Achse

Zweiter Hauptsatz der Thermodynamik

Kinematik des Punktes

Gerade ebene Balken

Eindimensionale Bewegung

Wirtschafts- und Finanzmathematik

Linearisierung nichtlinearer Systeme

Hier findest du Aufgaben mit Lösungen und Theorie zu: Linearkombination, Lineare Hüllen

Linearkombination, Lineare Hüllen | Aufgabensammlung mit Lösungen & Th