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الجبر الخطي الأمثلة
[1223-1-3322-346]⎡⎢⎣1223−1−3322−346⎤⎥⎦
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
خطوة 2.1
Perform the row operation R2=R2+R1R2=R2+R1 to make the entry at 2,12,1 a 00.
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+1⋅1-3+1⋅23+1⋅22+1⋅32-346]⎡⎢⎣1223−1+1⋅1−3+1⋅23+1⋅22+1⋅32−346⎤⎥⎦
خطوة 2.1.2
بسّط R2R2.
[12230-1552-346]⎡⎢⎣12230−1552−346⎤⎥⎦
[12230-1552-346]⎡⎢⎣12230−1552−346⎤⎥⎦
خطوة 2.2
Perform the row operation R3=R3-2R1R3=R3−2R1 to make the entry at 3,13,1 a 00.
خطوة 2.2.1
Perform the row operation R3=R3-2R1R3=R3−2R1 to make the entry at 3,13,1 a 00.
[12230-1552-2⋅1-3-2⋅24-2⋅26-2⋅3]⎡⎢⎣12230−1552−2⋅1−3−2⋅24−2⋅26−2⋅3⎤⎥⎦
خطوة 2.2.2
بسّط R3R3.
[12230-1550-700]⎡⎢⎣12230−1550−700⎤⎥⎦
[12230-1550-700]⎡⎢⎣12230−1550−700⎤⎥⎦
خطوة 2.3
Multiply each element of R2R2 by -1−1 to make the entry at 2,22,2 a 11.
خطوة 2.3.1
Multiply each element of R2R2 by -1−1 to make the entry at 2,22,2 a 11.
[1223-0--1-1⋅5-1⋅50-700]⎡⎢⎣1223−0−−1−1⋅5−1⋅50−700⎤⎥⎦
خطوة 2.3.2
بسّط R2R2.
[122301-5-50-700]⎡⎢⎣122301−5−50−700⎤⎥⎦
[122301-5-50-700]⎡⎢⎣122301−5−50−700⎤⎥⎦
خطوة 2.4
Perform the row operation R3=R3+7R2R3=R3+7R2 to make the entry at 3,23,2 a 00.
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+7⋅0-7+7⋅10+7⋅-50+7⋅-5]⎡⎢⎣122301−5−50+7⋅0−7+7⋅10+7⋅−50+7⋅−5⎤⎥⎦
خطوة 2.4.2
بسّط R3R3.
[122301-5-500-35-35]⎡⎢⎣122301−5−500−35−35⎤⎥⎦
[122301-5-500-35-35]⎡⎢⎣122301−5−500−35−35⎤⎥⎦
خطوة 2.5
Multiply each element of R3R3 by -135−135 to make the entry at 3,33,3 a 11.
خطوة 2.5.1
Multiply each element of R3R3 by -135−135 to make the entry at 3,33,3 a 11.
[122301-5-5-135⋅0-135⋅0-135⋅-35-135⋅-35]⎡⎢
⎢⎣122301−5−5−135⋅0−135⋅0−135⋅−35−135⋅−35⎤⎥
⎥⎦
خطوة 2.5.2
بسّط R3R3.
[122301-5-50011]⎡⎢⎣122301−5−50011⎤⎥⎦
[122301-5-50011]⎡⎢⎣122301−5−50011⎤⎥⎦
خطوة 2.6
Perform the row operation R2=R2+5R3R2=R2+5R3 to make the entry at 2,32,3 a 00.
خطوة 2.6.1
Perform the row operation R2=R2+5R3R2=R2+5R3 to make the entry at 2,32,3 a 00.
[12230+5⋅01+5⋅0-5+5⋅1-5+5⋅10011]⎡⎢⎣12230+5⋅01+5⋅0−5+5⋅1−5+5⋅10011⎤⎥⎦
خطوة 2.6.2
بسّط R2R2.
[122301000011]⎡⎢⎣122301000011⎤⎥⎦
[122301000011]⎡⎢⎣122301000011⎤⎥⎦
خطوة 2.7
Perform the row operation R1=R1-2R3R1=R1−2R3 to make the entry at 1,31,3 a 00.
خطوة 2.7.1
Perform the row operation R1=R1-2R3R1=R1−2R3 to make the entry at 1,31,3 a 00.
[1-2⋅02-2⋅02-2⋅13-2⋅101000011]⎡⎢⎣1−2⋅02−2⋅02−2⋅13−2⋅101000011⎤⎥⎦
خطوة 2.7.2
بسّط R1R1.
[120101000011]⎡⎢⎣120101000011⎤⎥⎦
[120101000011]⎡⎢⎣120101000011⎤⎥⎦
خطوة 2.8
Perform the row operation R1=R1-2R2R1=R1−2R2 to make the entry at 1,21,2 a 00.
خطوة 2.8.1
Perform the row operation R1=R1-2R2R1=R1−2R2 to make the entry at 1,21,2 a 00.
[1-2⋅02-2⋅10-2⋅01-2⋅001000011]⎡⎢⎣1−2⋅02−2⋅10−2⋅01−2⋅001000011⎤⎥⎦
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