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@ -1,91 +1,88 @@
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# Woche 12 # |
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A = [1, 2, -2, -1; 2, 0, -1, 1; 4, 3, 3, 1; 1, -2, 2, 3]; |
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## Quiz 11 ## |
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A eine m x m Matrix, m = 4: |
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Sei m = 4 und _A_ die folgende m x m Matrix über 𝔽₅: |
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A = 1 2 -2 -1 |
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2 0 -1 1 |
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4 3 3 1 |
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1 -2 2 3 |
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in 𝔽₅. |
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Zur Bestimmung der Invertierbarkeit: Gaußverfahren auf (A | I): |
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Zur Bestimmung der Invertierbarkeit führen wir das Gaußverfahren auf (A | I) aus: |
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1 2 -2 -1 | 1 0 0 0 |
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2 0 -1 1 | 0 1 0 0 |
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4 3 3 1 | 0 0 1 0 |
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1 -2 2 3 | 0 0 0 1 |
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1 2 -2 -1 | 1 0 0 0 |
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2 0 -1 1 | 0 1 0 0 |
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4 3 3 1 | 0 0 1 0 |
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1 -2 2 3 | 0 0 0 1 |
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Zeile 2 <- Zeile 2 - 2·Zeile 1 |
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Zeile 3 <- Zeile 3 - 4·Zeile 1 |
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Zeile 4 <- Zeile 4 - Zeile 1 |
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Zeile 2 <- Zeile 2 - 2·Zeile 1 |
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Zeile 3 <- Zeile 3 - 4·Zeile 1 |
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Zeile 4 <- Zeile 4 - Zeile 1 |
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1 2 -2 -1 | 1 0 0 0 |
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0 -4 3 3 | -2 1 0 0 |
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0 -5 11 5 | -4 0 1 0 |
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0 -4 4 4 | -1 0 0 1 |
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1 2 -2 -1 | 1 0 0 0 |
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0 -4 3 3 | -2 1 0 0 |
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0 -5 11 5 | -4 0 1 0 |
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0 -4 4 4 | -1 0 0 1 |
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—> modulo 5 |
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—> modulo 5 |
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1 2 3 4 | 1 0 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 1 4 4 | 4 0 0 1 |
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1 2 3 4 | 1 0 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 1 4 4 | 4 0 0 1 |
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Zeile 4 <- Zeile 4 - Zeile 2 |
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Zeile 4 <- Zeile 4 - Zeile 2 |
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1 2 3 4 | 1 0 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 1 1 | 1 4 0 1 |
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1 2 3 4 | 1 0 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 1 1 | 1 4 0 1 |
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(hier habe ich sofort mod 5 berechnet) |
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(hier habe ich sofort mod 5 berechnet) |
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Zeile 4 <- Zeile 4 - Zeile 3 |
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Zeile 4 <- Zeile 4 - Zeile 3 |
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1 2 3 4 | 1 0 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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1 2 3 4 | 1 0 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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⟹ Rang(A) = 4 = m |
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⟹ A invertierbar |
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⟹ _A_ ist invertierbar |
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Zeile 1 <- Zeile 1 - 2·Zeile 2 |
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Zeile 1 <- Zeile 1 - 2·Zeile 2 |
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1 0 2 3 | 0 3 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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1 0 2 3 | 0 3 0 0 |
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0 1 3 3 | 3 1 0 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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Zeile 1 <- Zeile 1 - 2·Zeile 3 |
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Zeile 2 <- Zeile 2 - 3·Zeile 3 |
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Zeile 1 <- Zeile 1 - 2·Zeile 3 |
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Zeile 2 <- Zeile 2 - 3·Zeile 3 |
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1 0 0 3 | 3 3 3 0 |
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0 1 0 3 | 0 1 2 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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1 0 0 3 | 3 3 3 0 |
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0 1 0 3 | 0 1 2 0 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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Zeile 1 <- Zeile 1 - 3·Zeile 4 |
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Zeile 2 <- Zeile 2 - 3·Zeile 4 |
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Zeile 1 <- Zeile 1 - 3·Zeile 4 |
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Zeile 2 <- Zeile 2 - 3·Zeile 4 |
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1 0 0 0 | 3 1 1 2 |
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0 1 0 0 | 0 4 0 2 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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1 0 0 0 | 3 1 1 2 |
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0 1 0 0 | 0 4 0 2 |
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0 0 1 0 | 1 0 1 0 |
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0 0 0 1 | 0 4 4 1 |
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⟹ A¯¹ steht in der rechten Hälfte |
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⟹ Das Produkt der Elementarmatrizen, die A auf I (linke Hälfte) reduziert hat, |
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steht nun in der rechten Hälfte: |
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A¯¹ = 3 1 1 2 |
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0 4 0 2 |
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1 0 1 0 |
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0 4 4 1 |
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A¯¹ = 3 1 1 2 |
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0 4 0 2 |
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1 0 1 0 |
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0 4 4 1 |
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## Lineare Ausdehnung ## |
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