Linearität einer Abbildung prüfen

Linearität einer Abbildung

Injektiv, Surjektiv, Bijektiv

Darstellungsmatrizen

Ungleichungen 

Statik in der Ebene

Fourier-Reihen

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Unterschied In/Homogen

Basis, Basiswechsel

Abbildungsmatrizen

Funktionen bestimmen

Geraden und Ebenen

Gruppen

lineare abbildung 

quadratische ergänzung

Baustatik 1

Anfangswertprobleme :)

Differentialgleichungen

basis 

QR-Algorithmus

QR-Zerlegung

Cholesky-Zerlegung

Bild & Urbild

Zweigstromanalyse

Transistor als Thermometer

Investment

Teilfolgen

Epsilon Delta Kriterium

prinzip der virtuellen verrückung 

Linear Kombination

Folgen, Reihen und Potenzreihen

Flächenpressung

Arbeit 

Cholesky Zerlegung

Polynomfunktionen beschreiben

Zu Maß-und Integrationstheorie: Majorisierte Konvergenz

Teilchenphysik und Festkörperphysik

Orthogonaltrajektorien

test

Taylorreihen

Regel von de L'Hopital

Mathe

KRAFT

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Folgenkriterium

ϵ-δ Kriterium 

Beweis Stetigkeit 

Wahrheitstabelle

2x2 Matrix

Matrix invertieren

Basiswechselsatz

Lineare Abbildunge 

Vorderrad fahradgabel lagerkräfte berechnen 

Spannungsregelung

Jordan Normal Form

Test

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Jordan-Normalform

Ansatz der Störfunktion zur Bestimmung der Partikulären Lösung

Kurvenintegral

dlg mithilfe inhomogenen linearen system lösen

Inverse einer Matrix

Gleichmäßige Stetigkeit

addition

Regel von L´Hospital

Lineare Abbildungen

dimension eines Untervektorraumes bestimmen

Stationäre Stoffübertragung

Polplan, Prinzip der virtuellen Verrückungen

komplexe Ungleichungen 

Bidualräume

Dualräume

Taylorpolynome

Kern und Basis

Diagonalisieren einer Matrix

Rekursive Folgen

Exponentialreihe

restglied

Basiswechsel

Gamma Funktion

Newton Verfahren

Randwertaufgabe

Numerische Integration: Trapez- und Simpsonregel

Funktionenfolgen

dualräume

Dualraum

Komplexe Nullstellen

Chemie

Natascha Kirchmann

Differentialquotient

Folgedefinition des Funktionslimes

Grenzwerte und Stetigkeit 

Komplexe Zahlen

Kern und Bild 

Direkte Summe 

<p>\begin{align*}_{B} M(f)_{A}=\left(\begin{array}{cc}</p>
<p>1 &amp; 0 \\</p>
<p>1 &amp; -3 \\</p>
<p>0 &amp; 1</p>
<p>\end{array}\right)\end{align*}</p>

<p>Stelle $f(a_1)$ als Linearkombination der Vektoren in $B$ dar.</p>


<p>Stelle <mjx-container class="MathJax" jax="SVG"><svg alt="f(a_1)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.114ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2260.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5442-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-5442-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-5442-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-5442-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-5442-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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-5442-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550, 0)"><use xlink:href="#MJX-5442-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(939, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-5442-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(529, -150) scale(0.707)"><use xlink:href="#MJX-5442-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1871.6, 0)"><use xlink:href="#MJX-5442-TEX-N-29"></use></g></g></g></svg></mjx-container> als Linearkombination der Vektoren in <mjx-container class="MathJax" jax="SVG"><svg alt="B" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5443-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></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-5443-TEX-I-1D435"></use></g></g></g></svg></mjx-container> dar.</p>


<p>\begin{align*}</p>
<p>f\left(a_{1}\right)&amp;=\left(\begin{array}{l}</p>
<p>3 \\1 \\1</p>
<p>\end{array}\right)\\</p>
<p>&amp;=\lambda_{1} b_{1}+\lambda_{2} b_{2}+\lambda_{3} b_{3}\\</p>
<p>&amp;=\lambda_{1}\left(\begin{array}{l}</p>
<p>1 \\0 \\1</p>
<p>\end{array}\right)+\lambda_{2}\left(\begin{array}{l}</p>
<p>2 \\1 \\0</p>
<p>\end{array}\right)+\lambda_{3}\left(\begin{array}{l}</p>
<p>1 \\1 \\0</p>
<p>\end{array}\right) </p>
<p>\end{align*}</p>


<p>Verwende das Gaußverfahren, um die $\lambda_i$ zu bestimmen. </p>


<p>Verwende das Gaußverfahren, um die <mjx-container class="MathJax" jax="SVG"><svg alt="\lambda_i" style="vertical-align: -0.357ex" xmlns="http://www.w3.org/2000/svg" width="1.984ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 877 851.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5445-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-5445-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></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="msub"><g data-mml-node="mi"><use xlink:href="#MJX-5445-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(583, -150) scale(0.707)"><use xlink:href="#MJX-5445-TEX-I-1D456"></use></g></g></g></g></svg></mjx-container> zu bestimmen. </p>


<p>\begin{align*}</p>
<p>\left(\begin{array}{ccc|c}</p>
<p>1 &amp; 2 &amp; 1 &amp; 3\\</p>
<p>0 &amp; 1 &amp; 1 &amp; 1\\</p>
<p>1 &amp; 0 &amp; 0 &amp; 1</p>
<p>\end{array}\right) </p>
<p>&amp;\rightarrow\left(\begin{array}{ccc|c}</p>
<p>1 &amp; 2 &amp; 1 &amp; 3\\</p>
<p>0 &amp; 1 &amp; 1 &amp; 1\\</p>
<p>0 &amp; -2 &amp; -1 &amp; -2</p>
<p>\end{array}\right) \\</p>
<p>&amp;\rightarrow\left(\begin{array}{lll|l}</p>
<p>1 &amp; 2 &amp; 1 &amp; 3 \\</p>
<p>0 &amp; 1 &amp; 1 &amp; 1\\</p>
<p>0 &amp; 0 &amp; 1 &amp; 0 </p>
<p>\end{array}\right)</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p> \Rightarrow \lambda_3&amp;=0</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow \lambda_2 &amp;=1- \lambda_3 \\&amp;= 1</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow \lambda_1 &amp;= 3-2\lambda_2-\lambda_3\\&amp;=3-2\\&amp;=1</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow f(a_1)=b_1+b_2</p>
<p>\end{align*}</p>


<p>Wie kannst du $f(a_2)$ berechnen?</p>


<p>Wie kannst du <mjx-container class="MathJax" jax="SVG"><svg alt="f(a_2)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.114ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2260.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5451-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-5451-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-5451-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-5451-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-5451-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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-5451-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550, 0)"><use xlink:href="#MJX-5451-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(939, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-5451-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(529, -150) scale(0.707)"><use xlink:href="#MJX-5451-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1871.6, 0)"><use xlink:href="#MJX-5451-TEX-N-29"></use></g></g></g></svg></mjx-container> berechnen?</p>


<p>Stelle $a_2$ als Linearkombination von $\left(\begin{array}{c}1 \\-1\end{array}\right)$ und $\left(\begin{array}{c}5 \\3\end{array}\right)$ dar:</p>


<p>Stelle <mjx-container class="MathJax" jax="SVG"><svg alt="a_2" style="vertical-align: -0.339ex" xmlns="http://www.w3.org/2000/svg" width="2.11ex" height="1.337ex" role="img" focusable="false" viewBox="0 -441 932.6 591" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5452-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-5452-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></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="msub"><g data-mml-node="mi"><use xlink:href="#MJX-5452-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(529, -150) scale(0.707)"><use xlink:href="#MJX-5452-TEX-N-32"></use></g></g></g></g></svg></mjx-container> als Linearkombination von <mjx-container class="MathJax" jax="SVG"><svg alt="\left(\begin{array}{c}1 \\-1\end{array}\right)" style="vertical-align: -2.149ex" xmlns="http://www.w3.org/2000/svg" width="6.222ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 2750 2400" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5453-TEX-S3-28" d="M701 -940Q701 -943 695 -949H664Q662 -947 636 -922T591 -879T537 -818T475 -737T412 -636T350 -511T295 -362T250 -186T221 17T209 251Q209 962 573 1361Q596 1386 616 1405T649 1437T664 1450H695Q701 1444 701 1441Q701 1436 681 1415T629 1356T557 1261T476 1118T400 927T340 675T308 359Q306 321 306 250Q306 -139 400 -430T690 -924Q701 -936 701 -940Z"></path><path id="MJX-5453-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-5453-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-5453-TEX-S3-29" d="M34 1438Q34 1446 37 1448T50 1450H56H71Q73 1448 99 1423T144 1380T198 1319T260 1238T323 1137T385 1013T440 864T485 688T514 485T526 251Q526 134 519 53Q472 -519 162 -860Q139 -885 119 -904T86 -936T71 -949H56Q43 -949 39 -947T34 -937Q88 -883 140 -813Q428 -430 428 251Q428 453 402 628T338 922T245 1146T145 1309T46 1425Q44 1427 42 1429T39 1433T36 1436L34 1438Z"></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="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-5453-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd" transform="translate(389, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-5453-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd"><g data-mml-node="mo"><use xlink:href="#MJX-5453-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-5453-TEX-N-31"></use></g></g></g></g><g data-mml-node="mo" transform="translate(2014, 0)"><use xlink:href="#MJX-5453-TEX-S3-29"></use></g></g></g></g></svg></mjx-container> und <mjx-container class="MathJax" jax="SVG"><svg alt="\left(\begin{array}{c}5 \\3\end{array}\right)" style="vertical-align: -2.149ex" xmlns="http://www.w3.org/2000/svg" width="4.462ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 1972 2400" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5454-TEX-S3-28" d="M701 -940Q701 -943 695 -949H664Q662 -947 636 -922T591 -879T537 -818T475 -737T412 -636T350 -511T295 -362T250 -186T221 17T209 251Q209 962 573 1361Q596 1386 616 1405T649 1437T664 1450H695Q701 1444 701 1441Q701 1436 681 1415T629 1356T557 1261T476 1118T400 927T340 675T308 359Q306 321 306 250Q306 -139 400 -430T690 -924Q701 -936 701 -940Z"></path><path id="MJX-5454-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-5454-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-5454-TEX-S3-29" d="M34 1438Q34 1446 37 1448T50 1450H56H71Q73 1448 99 1423T144 1380T198 1319T260 1238T323 1137T385 1013T440 864T485 688T514 485T526 251Q526 134 519 53Q472 -519 162 -860Q139 -885 119 -904T86 -936T71 -949H56Q43 -949 39 -947T34 -937Q88 -883 140 -813Q428 -430 428 251Q428 453 402 628T338 922T245 1146T145 1309T46 1425Q44 1427 42 1429T39 1433T36 1436L34 1438Z"></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="mrow"><g data-mml-node="mo"><use xlink:href="#MJX-5454-TEX-S3-28"></use></g><g data-mml-node="mtable" transform="translate(736, 0)"><g data-mml-node="mtr" transform="translate(0, 700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use xlink:href="#MJX-5454-TEX-N-35"></use></g></g></g><g data-mml-node="mtr" transform="translate(0, -700)"><g data-mml-node="mtd"><g data-mml-node="mn"><use xlink:href="#MJX-5454-TEX-N-33"></use></g></g></g></g><g data-mml-node="mo" transform="translate(1236, 0)"><use xlink:href="#MJX-5454-TEX-S3-29"></use></g></g></g></g></svg></mjx-container> dar:</p>


<p>\begin{align*}</p>
<p>\left(\begin{array}{l}1 \\3\end{array}\right)</p>
<p>=\mu_{1}\left(\begin{array}{c}1 \\-1\end{array}\right)</p>
<p>+\mu_{2}\left(\begin{array}{l}5 \\3\end{array}\right) </p>
<p>\end{align*}</p>


<p>Verwende dafür das Gaußverfahren. </p>


<p>\begin{align*}</p>
<p>\left(\begin{array}{cc|c}</p>
<p>1 &amp; 5 &amp; 1 \\</p>
<p>-1 &amp; 3 &amp; 3</p>
<p>\end{array}\right) \rightarrow\left(\begin{array}{ll|l}</p>
<p>1 &amp; 5 &amp; 1 \\</p>
<p>0 &amp; 8 &amp; 4</p>
<p>\end{array}\right)\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow \mu_2= \frac{1}{2}</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow \mu_1&amp;= 1-5\mu_2\\&amp;=-\frac{3}{2}</p>
<p>\end{align*}</p>


<p>Jetzt kannst du verwenden, dass $f$ eine lineare Abbildung ist und $f(a_2)$ ausrechnen. </p>


<p>Jetzt kannst du verwenden, dass <mjx-container class="MathJax" jax="SVG"><svg alt="f" style="vertical-align: -0.464ex" xmlns="http://www.w3.org/2000/svg" width="1.244ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 550 910" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5459-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></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-5459-TEX-I-1D453"></use></g></g></g></svg></mjx-container> eine lineare Abbildung ist und <mjx-container class="MathJax" jax="SVG"><svg alt="f(a_2)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.114ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2260.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5460-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-5460-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-5460-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-5460-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-5460-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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-5460-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550, 0)"><use xlink:href="#MJX-5460-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(939, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-5460-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(529, -150) scale(0.707)"><use xlink:href="#MJX-5460-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1871.6, 0)"><use xlink:href="#MJX-5460-TEX-N-29"></use></g></g></g></svg></mjx-container> ausrechnen. </p>


<p>\begin{align*}</p>
<p>f\left(\left(\begin{array}{l}1 \\3\end{array}\right)\right)</p>
<p>&amp;=f\left(\mu_{1}\left(\begin{array}{c}1 \\-1\end{array}\right)</p>
<p>+\mu_{2}\left(\begin{array}{l}5 \\3\end{array}\right)\right) </p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\phantom{f\left(\left(\begin{array}{l}1 \\3\end{array}\right)\right)}</p>
<p>&amp;=f\left(-\frac{3}{2}\left(\begin{array}{c}1 \\-1\end{array}\right)</p>
<p>+\frac{1}{2}\left(\begin{array}{l}5 \\3\end{array}\right)\right)</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\phantom{f\left(\left(\begin{array}{l}1 \\3\end{array}\right)\right)}</p>
<p>&amp;=-\frac{3}{2} f\left(\left(\begin{array}{c}1 \\-1\end{array}\right)\right)</p>
<p>+\frac{1}{2} f\left(\left(\begin{array}{l}5 \\3\end{array}\right)\right)</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\phantom{f\left(\left(\begin{array}{l}1 \\3\end{array}\right)\right)}</p>
<p>&amp;=-\frac{3}{2}\left(\begin{array}{l}3 \\1 \\1\end{array}\right)</p>
<p>+\frac{1}{2}\left(\begin{array}{c}-1 \\-1 \\3\end{array}\right)</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\phantom{f\left(\left(\begin{array}{l}1 \\3\end{array}\right)\right)}</p>
<p>&amp;=\left(\begin{array}{c}-5 \\-2 \\0\end{array}\right)</p>
<p>\end{align*}</p>


<p>Stelle $f(a_2)$ jetzt als Linearkombination der Vektoren in $B$ dar. </p>


<p>Stelle <mjx-container class="MathJax" jax="SVG"><svg alt="f(a_2)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.114ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2260.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5466-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-5466-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-5466-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-5466-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-5466-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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-5466-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550, 0)"><use xlink:href="#MJX-5466-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(939, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-5466-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(529, -150) scale(0.707)"><use xlink:href="#MJX-5466-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1871.6, 0)"><use xlink:href="#MJX-5466-TEX-N-29"></use></g></g></g></svg></mjx-container> jetzt als Linearkombination der Vektoren in <mjx-container class="MathJax" jax="SVG"><svg alt="B" style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5467-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></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-5467-TEX-I-1D435"></use></g></g></g></svg></mjx-container> dar. </p>


<p>\begin{align*}</p>
<p>f\left(a_{2}\right)&amp;=\left(\begin{array}{c}-5 \\-2 \\0\end{array}\right)\\</p>
<p>&amp;=\lambda_{1} b_{1}+\lambda_{2} b_{2}+\lambda_{3} b_{3}\\</p>
<p>&amp;=\lambda_{1}\left(\begin{array}{c}1 \\0 \\1\end{array}\right)</p>
<p>+\lambda_{2}\left(\begin{array}{l}2 \\1 \\0\end{array}\right)</p>
<p>+\lambda_{3}\left(\begin{array}{l}1 \\1 \\0\end{array}\right)</p>
<p>\end{align*}</p>


<p>Verwende das Gaußverfahren, um die <mjx-container class="MathJax" jax="SVG"><svg alt="\lambda_i" style="vertical-align: -0.357ex" xmlns="http://www.w3.org/2000/svg" width="1.984ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 877 851.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5469-TEX-I-1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path id="MJX-5469-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></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="msub"><g data-mml-node="mi"><use xlink:href="#MJX-5469-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(583, -150) scale(0.707)"><use xlink:href="#MJX-5469-TEX-I-1D456"></use></g></g></g></g></svg></mjx-container> zu bestimmen. </p>


<p>\begin{align*}</p>
<p>\left(\begin{array}{lll|l}</p>
<p>1 &amp; 2 &amp; 1 &amp; -5 \\</p>
<p>0 &amp; 1 &amp; 1 &amp; -2 \\</p>
<p>1 &amp; 0 &amp; 0 &amp; 0</p>
<p>\end{array}\right)</p>
<p>&amp;\rightarrow\left(\begin{array}{ccc|c}</p>
<p>1 &amp; 2 &amp; 1 &amp; -5 \\</p>
<p>0 &amp; 1 &amp; 1 &amp; -2 \\</p>
<p>0 &amp; -2 &amp; -1 &amp; 5</p>
<p>\end{array}\right) \\</p>
<p>&amp;\rightarrow\left(\begin{array}{ccc|c}</p>
<p>1 &amp; 2 &amp; 1 &amp; -5 \\</p>
<p>0 &amp; 1 &amp; 1 &amp; -2 \\</p>
<p>0 &amp; 0 &amp; 1 &amp; 1</p>
<p>\end{array}\right)</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p> \Rightarrow \lambda_3&amp;=1</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow \lambda_2 &amp;=-2-\lambda_1 \\&amp;= -3</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow \lambda_1 &amp;= -5-2\lambda_2-\lambda_3\\&amp;=-5+6-1\\&amp;=0</p>
<p>\end{align*}</p>


<p>\begin{align*}</p>
<p>\Rightarrow f(a_2)=-3b_2+b_3</p>
<p>\end{align*}</p>

<p>Sei $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ eine lineare Abbildung. Weiter seien</p>
<p>\begin{align*}</p>
<p>A&amp;=\left\{a_{1}, a_{2}\right\}=\left\{\left(\begin{array}{c}</p>
<p>1 \\-1</p>
<p>\end{array}\right),\left(\begin{array}{l}</p>
<p>1 \\3</p>
<p>\end{array}\right)\right\} \quad \text { und } \\\quad B&amp;=\left\{b_{1}, b_{2}, b_{3}\right\}=\left\{\left(\begin{array}{l}</p>
<p>1 \\</p>
<p>0 \\</p>
<p>1</p>
<p>\end{array}\right),\left(\begin{array}{l}</p>
<p>2 \\</p>
<p>1 \\</p>
<p>0</p>
<p>\end{array}\right),\left(\begin{array}{l}</p>
<p>1 \\</p>
<p>1 \\</p>
<p>0</p>
<p>\end{array}\right)\right\}</p>
<p>\end{align*}</p>
<p>geordnete Basen des $\mathbb{R}^{2}$ bzw. des $\mathbb{R}^{3}$. Zusätzlich gelte</p>
<p>\begin{align*}</p>
<p>f\left(\left(\begin{array}{c}</p>
<p>1 \\-1</p>
<p>\end{array}\right)\right)=\left(\begin{array}{l}</p>
<p>3 \\1 \\1</p>
<p>\end{array}\right) \quad \text { und } \quad f\left(\left(\begin{array}{l}</p>
<p>5 \\3</p>
<p>\end{array}\right)\right)=\left(\begin{array}{c}</p>
<p>-1 \\-1 \\3</p>
<p>\end{array}\right).</p>
<p>\end{align*}</p>
<p></p>
<p>Bestimme die Darstellungsmatrix $_{B} M(f)_{A}$.</p>

Dies ist eine interaktive Aufgabe zu: Darstellungsmatrizen mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Aufgabenstellung:

Lösungsweg:

Lösung: