Explain in detailed steps b) The coefficients of the following system are taken from GF(2). Solve it using Gaus-sian elimination.X1 + X2 + X4 = 0X1 + X3 + X4 =1X2 + X5 = 0X1 + X2 + X3 + x5 = 0X1 + X3 = 0(b) Gaussian elimination in compressed form should look somewhat like the following.1 1 0 1 0|01 0 1 1 010 1 0 0 101 1 10 10\1 0 1 0 0|01 1 0 1 0|00 1 1 0 0 10 1 0 0 100 0 1 1 1 00 1 1 1 0|0add row 1 to rows 2,4,51 1 0 1 0 00 1 10 0|10 0 1 0 1 10 0 1 1 100 0 0 1 0 1add row 2 to rows 3,51 1 0 10|00 1 1 0 0 10 0 1 0 1 10 0 0 1 0|10 0 0 1 0 1row 4 + row 3 + row 41 1 0 1 0 00 1 1000 0 1 0 1 |10 0 0 1 0 10 0 0 0 0|0row 5 + row 5 + row 41 1 0 1 0|00 1 0 0 100 0 10 1|10 0 0 10 10 0 0 0 0 0row 2 + row 2 + row 31 0 0 1 1|00 1 0 0 100 0 1 0 110 0 0 1 0|10 0 0 0 0|0row 1 + row 1 + row 20 110 1 0 0 100 0 10 110 0 0 1 0 |10 0 0 0 001row 1 + row 1 + row 4This shows xs is free. As GF(2) only has two elements (0 and 1), the solutions areX5 = 0,X4 = 1,X3 = 1,X2 = 0,X1 = 1andX5 = 1,X4 = 1,X3 = 0, X2 = 1,X1 = 0.

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Answered: b) The coefficients of the following…

b) The coefficients of the following system are taken from GF(2). Solve it using Gaus- sian elimination. X1 + x2 + x4 = 0 X1 + X3 + X4 = 1 X2 + X5 = 0 X1 + x2 + x3 + X5 = 0 X1 + x3 = 0

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.2: Direct Methods For Solving Linear Systems

Problem 31EQ: In Exercises 25-34, solve the given system of equations using either Gaussian or Gauss-Jordan...

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Question

Explain in detailed steps

b) The coefficients of the following system are taken from GF(2). Solve it using Gaus-
sian elimination.
X1 + X2 + X4 = 0
X1 + X3 + X4 =1
X2 + X5 = 0
X1 + X2 + X3 + x5 = 0
X1 + X3 = 0
(b) Gaussian elimination in compressed form should look somewhat like the following.
1 1 0 1 0|0
1 0 1 1 01
0 1 0 0 10
1 1 10 10
\1 0 1 0 0|0
1 1 0 1 0|0
0 1 1 0 0 1
0 1 0 0 10
0 0 1 1 1 0
0 1 1 1 0|0
add row 1 to rows 2,4,5
1 1 0 1 0 0
0 1 10 0|1
0 0 1 0 1 1
0 0 1 1 10
0 0 0 1 0 1
add row 2 to rows 3,5
1 1 0 10|0
0 1 1 0 0 1
0 0 1 0 1 1
0 0 0 1 0|1
0 0 0 1 0 1
row 4 + row 3 + row 4
1 1 0 1 0 0
0 1 100
0 0 1 0 1 |1
0 0 0 1 0 1
0 0 0 0 0|0
row 5 + row 5 + row 4
1 1 0 1 0|0
0 1 0 0 10
0 0 10 1|1
0 0 0 10 1
0 0 0 0 0 0
row 2 + row 2 + row 3
1 0 0 1 1|0
0 1 0 0 10
0 0 1 0 11
0 0 0 1 0|1
0 0 0 0 0|0
row 1 + row 1 + row 2
0 11
0 1 0 0 10
0 0 10 11
0 0 0 1 0 |1
0 0 0 0 00
1
row 1 + row 1 + row 4
This shows xs is free. As GF(2) only has two elements (0 and 1), the solutions are
X5 = 0,
X4 = 1,
X3 = 1,
X2 = 0,
X1 = 1
and
X5 = 1,
X4 = 1,
X3 = 0, X2 = 1,
X1 = 0.

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