الجبر الخطي الأمثلة

أوجد الصفرية [[1,2,2,3],[-1,-3,3,2],[2,-3,4,6]]
[1223-1-3322-346]122313322346
خطوة 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.
خطوة 2
أوجِد الصيغة الدرجية المختزلة صفيًا.
انقر لعرض المزيد من الخطوات...
خطوة 2.1
Perform the row operation R2=R2+R1R2=R2+R1 to make the entry at 2,12,1 a 00.
انقر لعرض المزيد من الخطوات...
خطوة 2.1.1
Perform the row operation R2=R2+R1R2=R2+R1 to make the entry at 2,12,1 a 00.
[1223-1+11-3+123+122+132-346]12231+113+123+122+132346
خطوة 2.1.2
بسّط R2R2.
[12230-1552-346]122301552346
[12230-1552-346]122301552346
خطوة 2.2
Perform the row operation R3=R3-2R1R3=R32R1 to make the entry at 3,13,1 a 00.
انقر لعرض المزيد من الخطوات...
خطوة 2.2.1
Perform the row operation R3=R3-2R1R3=R32R1 to make the entry at 3,13,1 a 00.
[12230-1552-21-3-224-226-23]12230155221322422623
خطوة 2.2.2
بسّط R3R3.
[12230-1550-700]122301550700
[12230-1550-700]122301550700
خطوة 2.3
Multiply each element of R2R2 by -11 to make the entry at 2,22,2 a 11.
انقر لعرض المزيد من الخطوات...
خطوة 2.3.1
Multiply each element of R2R2 by -11 to make the entry at 2,22,2 a 11.
[1223-0--1-15-150-700]12230115150700
خطوة 2.3.2
بسّط R2R2.
[122301-5-50-700]122301550700
[122301-5-50-700]122301550700
خطوة 2.4
Perform the row operation R3=R3+7R2R3=R3+7R2 to make the entry at 3,23,2 a 00.
انقر لعرض المزيد من الخطوات...
خطوة 2.4.1
Perform the row operation R3=R3+7R2R3=R3+7R2 to make the entry at 3,23,2 a 00.
[122301-5-50+70-7+710+7-50+7-5]122301550+707+710+750+75
خطوة 2.4.2
بسّط R3R3.
[122301-5-500-35-35]12230155003535
[122301-5-500-35-35]12230155003535
خطوة 2.5
Multiply each element of R3R3 by -135135 to make the entry at 3,33,3 a 11.
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خطوة 2.5.1
Multiply each element of R3R3 by -135135 to make the entry at 3,33,3 a 11.
[122301-5-5-1350-1350-135-35-135-35]⎢ ⎢12230155135013501353513535⎥ ⎥
خطوة 2.5.2
بسّط R3R3.
[122301-5-50011]122301550011
[122301-5-50011]122301550011
خطوة 2.6
Perform the row operation R2=R2+5R3R2=R2+5R3 to make the entry at 2,32,3 a 00.
انقر لعرض المزيد من الخطوات...
خطوة 2.6.1
Perform the row operation R2=R2+5R3R2=R2+5R3 to make the entry at 2,32,3 a 00.
[12230+501+50-5+51-5+510011]12230+501+505+515+510011
خطوة 2.6.2
بسّط R2R2.
[122301000011]122301000011
[122301000011]122301000011
خطوة 2.7
Perform the row operation R1=R1-2R3R1=R12R3 to make the entry at 1,31,3 a 00.
انقر لعرض المزيد من الخطوات...
خطوة 2.7.1
Perform the row operation R1=R1-2R3R1=R12R3 to make the entry at 1,31,3 a 00.
[1-202-202-213-2101000011]12022022132101000011
خطوة 2.7.2
بسّط R1R1.
[120101000011]120101000011
[120101000011]120101000011
خطوة 2.8
Perform the row operation R1=R1-2R2R1=R12R2 to make the entry at 1,21,2 a 00.
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خطوة 2.8.1
Perform the row operation R1=R1-2R2R1=R12R2 to make the entry at 1,21,2 a 00.
[1-202-210-201-2001000011]12022102012001000011
خطوة 2.8.2
بسّط R1R1.
[100101000011]100101000011
[100101000011]100101000011
[100101000011]100101000011
خطوة 3
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
خطوة 4
The nullity is the number of columns without a pivot position in the row reduced matrix.
11
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