Solve the system 3 x+9 y+4 z=8 2 x+3 y+5 z=2 5 x+4 y+9 z=10 over G F11.

Name: Problem 5, Chapter 9: Applied Algebra: Codes, Ciphers and Discrete Algorithms
Uploaded: 2020-07-01T10:18:36-08:00
Duration: 3 min 27 s
Channel: Suman Saurav Thakur
Description: Solve the system 3 x+9 y+4 z=8 2 x+3 y+5 z=2 5 x+4 y+9 z=10 over G F11.

step 1 We have problem number 10 in which we need to solve the simultaneous linear equations, three sim

Solve the system 3 x+9 y+4 z=8 2 x+3 y+5 z=2 ...

Key Concepts

Systems of Linear Equations

A system of linear equations is a set of equations involving the same set of variables. The goal is to find values for the variables that simultaneously satisfy all the equations. When the system is defined over GF(11), the solution process must respect the modular arithmetic rules of the finite field.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by transforming the coefficient matrix into row echelon form through a sequence of row operations. When applied over a finite field like GF(11), the same procedure is followed, with all arithmetic calculations performed modulo 11.

Finite Fields

A finite field is a field with a finite number of elements. In the case of GF(11), the elements are the integers from 0 to 10, with addition and multiplication performed modulo 11. Every nonzero element in a finite field has a multiplicative inverse, making it possible to perform division operations within the field.

Modular Arithmetic

Modular arithmetic involves performing arithmetic operations with integers where the results are taken modulo a given number. In problems over GF(11), addition, subtraction, multiplication, and division (using modular inverses) are all carried out modulo 11, affecting how equations are simplified and solved.

Modular Inverses

Modular inverses are used to 'divide' by a number in modular arithmetic. For a number a (not zero) in GF(11), the modular inverse is an element b such that a * b is congruent to 1 modulo 11. This concept is essential when performing row operations in Gaussian elimination, as it allows for the normalization of pivot entries.

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