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الجبر الخطي الأمثلة
[14030001-300000-1]⎡⎢⎣14030001−300000−1⎤⎥⎦
خطوة 1
Write as an augmented matrix for Ax=0Ax=0.
[140300001-3000000-10]⎡⎢
⎢⎣140300001−3000000−10⎤⎥
⎥⎦
خطوة 2
خطوة 2.1
Multiply each element of R3R3 by -1−1 to make the entry at 3,53,5 a 11.
خطوة 2.1.1
Multiply each element of R3R3 by -1−1 to make the entry at 3,53,5 a 11.
[140300001-300-0-0-0-0--1-0]⎡⎢
⎢⎣140300001−300−0−0−0−0−−1−0⎤⎥
⎥⎦
خطوة 2.1.2
بسّط R3R3.
[140300001-300000010]⎡⎢
⎢⎣140300001−300000010⎤⎥
⎥⎦
[140300001-300000010]⎡⎢
⎢⎣140300001−300000010⎤⎥
⎥⎦
[140300001-300000010]⎡⎢
⎢⎣140300001−300000010⎤⎥
⎥⎦
خطوة 3
Use the result matrix to declare the final solution to the system of equations.
x1+4x2+3x4=0x1+4x2+3x4=0
x3-3x4=0x3−3x4=0
x5=0x5=0
خطوة 4
Write a solution vector by solving in terms of the free variables in each row.
[x1x2x3x4x5]=[-4x2-3x4x23x4x40]⎡⎢
⎢
⎢
⎢
⎢
⎢⎣x1x2x3x4x5⎤⎥
⎥
⎥
⎥
⎥
⎥⎦=⎡⎢
⎢
⎢
⎢
⎢
⎢⎣−4x2−3x4x23x4x40⎤⎥
⎥
⎥
⎥
⎥
⎥⎦
خطوة 5
Write the solution as a linear combination of vectors.
[x1x2x3x4x5]=x2[-41000]+x4[-30310]⎡⎢
⎢
⎢
⎢
⎢
⎢⎣x1x2x3x4x5⎤⎥
⎥
⎥
⎥
⎥
⎥⎦=x2⎡⎢
⎢
⎢
⎢
⎢
⎢⎣−41000⎤⎥
⎥
⎥
⎥
⎥
⎥⎦+x4⎡⎢
⎢
⎢
⎢
⎢
⎢⎣−30310⎤⎥
⎥
⎥
⎥
⎥
⎥⎦
خطوة 6
Write as a solution set.
{x2[-41000]+x4[-30310]|x2,x4∈R}⎧⎪
⎪
⎪
⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩x2⎡⎢
⎢
⎢
⎢
⎢
⎢⎣−41000⎤⎥
⎥
⎥
⎥
⎥
⎥⎦+x4⎡⎢
⎢
⎢
⎢
⎢
⎢⎣−30310⎤⎥
⎥
⎥
⎥
⎥
⎥⎦∣∣
∣
∣
∣
∣
∣∣x2,x4∈R⎫⎪
⎪
⎪
⎪
⎪
⎪
⎪⎬⎪
⎪
⎪
⎪
⎪
⎪
⎪⎭