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

Wirkungsplan

(Matrix-) Normen berechnen

elektrische Systeme

Kaskaden-Regelung

Vektoranalysis und Mehrfachintegrale

Diskretisierung

Mehrfachintegrale berechnen

Übertragungsfunktionen 

Ziegler-Nichols

Satz von Gauß

Zustandsregler

Satz von Stokes

Stochastik

Satz von Green

Grundlagen

Informatik

Summenformel

Algorithmen

TU München

Sortieralgorithmen

Linearität einer Abbildung prüfen

Quicksort

Implizite Funktionen

mechanische Systeme 

Ortskurve

Metrische Räume

Wärmestrahlung

Beweisaufgabe

Lichtquanten

thermische Systeme 

Materiewellen

Reibung (Gleitreibung)

Atom- und Kernphysik

(Weg)zusammenhängende Räume

Schwerpunkte paralleler angreifender Kräfte

Wahr/Falsch

Unschärferelation

Fundamentalgruppen

Resultierende Kräfte und Momente ermitteln

Quantenbedingungen nach Bohr/Sommerfeld

Mannigfaltigkeiten

Schrödinger-Gleichung

Tunneleffekt

(Lokal) Kompakte Räume

Resultierende und Zerlegung von Kräften

Stetigkeit (mehrdimensional)

Funktionen (mehrdimensionale)

Kompakt oder nicht?

Kraft- und Drehmomente

Haftung (Haftreibung)

Optisches Spektrum des Wasserstoffatoms

Vektorbündel

Bohr’sche Postulate

Gerade ebene Balken

Quantenmechanik des Wasserstoffatoms

Produkt- und Quotientenräume

Zeeman- und Stark-Effekt

Spin-Bahnkopplung

Elektrotechnik

Statische und kinematische Bestimmtheit (Fachwerke)

Gleichstrom

Triviale DGL

Zustandsraumdarstellung

Elektronen- und Kernspinresonanz

Stabilität

Aufbau der Elektronenhülle

Trennung der Variablen (Separation)

Ladung, Strom, Stromdichte, Spannung

Radioaktiver Zerfall

Absorption von Röntgenstrahlung, (Computertomografie)

Hurwitz-Kriterium

 Rotations-Schwingungs-Spektrum

Routh-Kriterium

Modellierung

Elektrische Feldgrößen 

Thermodynamik

Regelungsentwurf

Thermodynamische Grundbegriffe

zeitdiskrete und kontinuierliche Systeme

Thermische Ausdehnung

dynamische Systeme im Bildbereich 

Systemgestaltung 

Allgemeine Zustandsgleichung idealer Gase

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

Linearkombination, Lineare Hüllen | Aufgabensammlung mit Lösungen & Th