الجبر الخطي الأمثلة

أوجد فضاء العينة [[0,4,-3,-1],[4,4,-2,-2],[-1,-1,1,-1],[0,2,1,3]]
[04-3-144-2-2-1-11-10213]
خطوة 1
Write as an augmented matrix for Ax=0.
[04-3-1044-2-20-1-11-1002130]
خطوة 2
أوجِد الصيغة الدرجية المختزلة صفيًا.
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خطوة 2.1
Swap R2 with R1 to put a nonzero entry at 1,1.
[44-2-2004-3-10-1-11-1002130]
خطوة 2.2
Multiply each element of R1 by 14 to make the entry at 1,1 a 1.
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خطوة 2.2.1
Multiply each element of R1 by 14 to make the entry at 1,1 a 1.
[4444-24-240404-3-10-1-11-1002130]
خطوة 2.2.2
بسّط R1.
[11-12-12004-3-10-1-11-1002130]
[11-12-12004-3-10-1-11-1002130]
خطوة 2.3
Perform the row operation R3=R3+R1 to make the entry at 3,1 a 0.
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خطوة 2.3.1
Perform the row operation R3=R3+R1 to make the entry at 3,1 a 0.
[11-12-12004-3-10-1+11-1+111-12-1-120+002130]
خطوة 2.3.2
بسّط R3.
[11-12-12004-3-100012-32002130]
[11-12-12004-3-100012-32002130]
خطوة 2.4
Multiply each element of R2 by 14 to make the entry at 2,2 a 1.
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خطوة 2.4.1
Multiply each element of R2 by 14 to make the entry at 2,2 a 1.
[11-12-1200444-34-14040012-32002130]
خطوة 2.4.2
بسّط R2.
[11-12-12001-34-1400012-32002130]
[11-12-12001-34-1400012-32002130]
خطوة 2.5
Perform the row operation R4=R4-2R2 to make the entry at 4,2 a 0.
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خطوة 2.5.1
Perform the row operation R4=R4-2R2 to make the entry at 4,2 a 0.
[11-12-12001-34-1400012-3200-202-211-2(-34)3-2(-14)0-20]
خطوة 2.5.2
بسّط R4.
[11-12-12001-34-1400012-3200052720]
[11-12-12001-34-1400012-3200052720]
خطوة 2.6
Multiply each element of R3 by 2 to make the entry at 3,3 a 1.
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خطوة 2.6.1
Multiply each element of R3 by 2 to make the entry at 3,3 a 1.
[11-12-12001-34-14020202(12)2(-32)200052720]
خطوة 2.6.2
بسّط R3.
[11-12-12001-34-140001-300052720]
[11-12-12001-34-140001-300052720]
خطوة 2.7
Perform the row operation R4=R4-52R3 to make the entry at 4,3 a 0.
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خطوة 2.7.1
Perform the row operation R4=R4-52R3 to make the entry at 4,3 a 0.
[11-12-12001-34-140001-300-5200-52052-52172-52-30-520]
خطوة 2.7.2
بسّط R4.
[11-12-12001-34-140001-30000110]
[11-12-12001-34-140001-30000110]
خطوة 2.8
Multiply each element of R4 by 111 to make the entry at 4,4 a 1.
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خطوة 2.8.1
Multiply each element of R4 by 111 to make the entry at 4,4 a 1.
[11-12-12001-34-140001-300110110111111011]
خطوة 2.8.2
بسّط R4.
[11-12-12001-34-140001-3000010]
[11-12-12001-34-140001-3000010]
خطوة 2.9
Perform the row operation R3=R3+3R4 to make the entry at 3,4 a 0.
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خطوة 2.9.1
Perform the row operation R3=R3+3R4 to make the entry at 3,4 a 0.
[11-12-12001-34-1400+300+301+30-3+310+3000010]
خطوة 2.9.2
بسّط R3.
[11-12-12001-34-1400010000010]
[11-12-12001-34-1400010000010]
خطوة 2.10
Perform the row operation R2=R2+14R4 to make the entry at 2,4 a 0.
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خطوة 2.10.1
Perform the row operation R2=R2+14R4 to make the entry at 2,4 a 0.
[11-12-1200+1401+140-34+140-14+1410+1400010000010]
خطوة 2.10.2
بسّط R2.
[11-12-12001-34000010000010]
[11-12-12001-34000010000010]
خطوة 2.11
Perform the row operation R1=R1+12R4 to make the entry at 1,4 a 0.
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خطوة 2.11.1
Perform the row operation R1=R1+12R4 to make the entry at 1,4 a 0.
[1+1201+120-12+120-12+1210+12001-34000010000010]
خطوة 2.11.2
بسّط R1.
[11-120001-34000010000010]
[11-120001-34000010000010]
خطوة 2.12
Perform the row operation R2=R2+34R3 to make the entry at 2,3 a 0.
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خطوة 2.12.1
Perform the row operation R2=R2+34R3 to make the entry at 2,3 a 0.
[11-12000+3401+340-34+3410+3400+3400010000010]
خطوة 2.12.2
بسّط R2.
[11-1200010000010000010]
[11-1200010000010000010]
خطوة 2.13
Perform the row operation R1=R1+12R3 to make the entry at 1,3 a 0.
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خطوة 2.13.1
Perform the row operation R1=R1+12R3 to make the entry at 1,3 a 0.
[1+1201+120-12+1210+1200+120010000010000010]
خطوة 2.13.2
بسّط R1.
[11000010000010000010]
[11000010000010000010]
خطوة 2.14
Perform the row operation R1=R1-R2 to make the entry at 1,2 a 0.
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خطوة 2.14.1
Perform the row operation R1=R1-R2 to make the entry at 1,2 a 0.
[1-01-10-00-00-0010000010000010]
خطوة 2.14.2
بسّط R1.
[10000010000010000010]
[10000010000010000010]
[10000010000010000010]
خطوة 3
Use the result matrix to declare the final solution to the system of equations.
x1=0
x2=0
x3=0
x4=0
خطوة 4
Write a solution vector by solving in terms of the free variables in each row.
[x1x2x3x4]=[0000]
خطوة 5
Write as a solution set.
{[0000]}
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