Lineare Algebra Beispiele

Ermittle die Pivot-Positionen und -Spalten
[24681012120]24681012120
Schritt 1
Ermittele die normierte Zeilenstufenform.
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Schritt 1.1
Multiply each element of R1R1 by 1212 to make the entry at 1,11,1 a 11.
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Schritt 1.1.1
Multiply each element of R1R1 by 1212 to make the entry at 1,11,1 a 11.
[22426281012120]⎢ ⎢22426281012120⎥ ⎥
Schritt 1.1.2
Vereinfache R1R1.
[12381012120]12381012120
[12381012120]12381012120
Schritt 1.2
Perform the row operation R2=R2-8R1R2=R28R1 to make the entry at 2,12,1 a 00.
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Schritt 1.2.1
Perform the row operation R2=R2-8R1R2=R28R1 to make the entry at 2,12,1 a 00.
[1238-8110-8212-83120]12388110821283120
Schritt 1.2.2
Vereinfache R2R2.
[1230-6-12120]1230612120
[1230-6-12120]1230612120
Schritt 1.3
Perform the row operation R3=R3-R1R3=R3R1 to make the entry at 3,13,1 a 00.
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Schritt 1.3.1
Perform the row operation R3=R3-R1R3=R3R1 to make the entry at 3,13,1 a 00.
[1230-6-121-12-20-3]1230612112203
Schritt 1.3.2
Vereinfache R3R3.
[1230-6-1200-3]1230612003
[1230-6-1200-3]1230612003
Schritt 1.4
Multiply each element of R2R2 by -1616 to make the entry at 2,22,2 a 11.
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Schritt 1.4.1
Multiply each element of R2R2 by -1616 to make the entry at 2,22,2 a 11.
[123-160-16-6-16-1200-3]⎢ ⎢1231601661612003⎥ ⎥
Schritt 1.4.2
Vereinfache R2R2.
[12301200-3]123012003
[12301200-3]123012003
Schritt 1.5
Multiply each element of R3R3 by -1313 to make the entry at 3,33,3 a 11.
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Schritt 1.5.1
Multiply each element of R3R3 by -1313 to make the entry at 3,33,3 a 11.
[123012-130-130-13-3]⎢ ⎢123012130130133⎥ ⎥
Schritt 1.5.2
Vereinfache R3R3.
[123012001]123012001
[123012001]123012001
Schritt 1.6
Perform the row operation R2=R2-2R3R2=R22R3 to make the entry at 2,32,3 a 00.
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Schritt 1.6.1
Perform the row operation R2=R2-2R3R2=R22R3 to make the entry at 2,32,3 a 00.
[1230-201-202-21001]123020120221001
Schritt 1.6.2
Vereinfache R2R2.
[123010001]123010001
[123010001]123010001
Schritt 1.7
Perform the row operation R1=R1-3R3R1=R13R3 to make the entry at 1,31,3 a 00.
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Schritt 1.7.1
Perform the row operation R1=R1-3R3R1=R13R3 to make the entry at 1,31,3 a 00.
[1-302-303-31010001]130230331010001
Schritt 1.7.2
Vereinfache R1R1.
[120010001]120010001
[120010001]120010001
Schritt 1.8
Perform the row operation R1=R1-2R2R1=R12R2 to make the entry at 1,21,2 a 00.
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Schritt 1.8.1
Perform the row operation R1=R1-2R2R1=R12R2 to make the entry at 1,21,2 a 00.
[1-202-210-20010001]120221020010001
Schritt 1.8.2
Vereinfache R1R1.
[100010001]100010001
[100010001]100010001
[100010001]100010001
Schritt 2
The pivot positions are the locations with the leading 11 in each row. The pivot columns are the columns that have a pivot position.
Pivot Positions: a11,a22,a11,a22, and a33a33
Pivot Columns: 1,2,1,2, and 33
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 [x2  12  π  xdx ]  x2  12  π  xdx  
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