Beispiele
[4045678]⎡⎢
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⎢⎣4045678⎤⎥
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Schritt 1
Nullity is the dimension of the null space, which is the same as the number of free variables in the system after row reducing. The free variables are the columns without pivot positions.
Schritt 2
Schritt 2.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
Schritt 2.1.1
Multiply each element of R1R1 by 1414 to make the entry at 1,11,1 a 11.
[44045678]⎡⎢
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⎢⎣44045678⎤⎥
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Schritt 2.1.2
Vereinfache R1R1.
[1045678]⎡⎢
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[1045678]⎡⎢
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⎢⎣1045678⎤⎥
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Schritt 2.2
Perform the row operation R3=R3-4R1R3=R3−4R1 to make the entry at 3,13,1 a 00.
Schritt 2.2.1
Perform the row operation R3=R3-4R1R3=R3−4R1 to make the entry at 3,13,1 a 00.
[104-4⋅15678]⎡⎢
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⎢⎣104−4⋅15678⎤⎥
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Schritt 2.2.2
Vereinfache R3R3.
[1005678]⎡⎢
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⎢⎣1005678⎤⎥
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[1005678]⎡⎢
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⎢⎣1005678⎤⎥
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Schritt 2.3
Perform the row operation R4=R4-5R1R4=R4−5R1 to make the entry at 4,14,1 a 00.
Schritt 2.3.1
Perform the row operation R4=R4-5R1R4=R4−5R1 to make the entry at 4,14,1 a 00.
[1005-5⋅1678]⎡⎢
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⎢⎣1005−5⋅1678⎤⎥
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Schritt 2.3.2
Vereinfache R4R4.
[1000678]⎡⎢
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⎢⎣1000678⎤⎥
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[1000678]⎡⎢
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⎢⎣1000678⎤⎥
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Schritt 2.4
Perform the row operation R5=R5-6R1R5=R5−6R1 to make the entry at 5,15,1 a 00.
Schritt 2.4.1
Perform the row operation R5=R5-6R1R5=R5−6R1 to make the entry at 5,15,1 a 00.
[10006-6⋅178]⎡⎢
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⎢⎣10006−6⋅178⎤⎥
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Schritt 2.4.2
Vereinfache R5R5.
[1000078]⎡⎢
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⎢⎣1000078⎤⎥
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[1000078]⎡⎢
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⎢⎣1000078⎤⎥
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Schritt 2.5
Perform the row operation R6=R6-7R1 to make the entry at 6,1 a 0.
Schritt 2.5.1
Perform the row operation R6=R6-7R1 to make the entry at 6,1 a 0.
[100007-7⋅18]
Schritt 2.5.2
Vereinfache R6.
[1000008]
[1000008]
Schritt 2.6
Perform the row operation R7=R7-8R1 to make the entry at 7,1 a 0.
Schritt 2.6.1
Perform the row operation R7=R7-8R1 to make the entry at 7,1 a 0.
[1000008-8⋅1]
Schritt 2.6.2
Vereinfache R7.
[1000000]
[1000000]
[1000000]
Schritt 3
The pivot positions are the locations with the leading 1 in each row. The pivot columns are the columns that have a pivot position.
Pivot Positions: a11
Pivot Columns: 1
Schritt 4
The nullity is the number of columns without a pivot position in the row reduced matrix.
0