Question

Consider the linear system \(3 x+y+4 z=3\). In each case solve the system by \(4 x+3 y+z=1\) reducing the augmented matrix to reduced row-echelon form over the given field: a. \(\mathbb{Z}_{5}\) b. \(\mathbb{Z}_{7}\)

Short answer

\(\mathbb{Z}_5\): \((x, y, z) = (3 - 2y, y, 1)\); \(\mathbb{Z}_7\): \((x, y, z) = (3 - 4z, 6 - 2z, z)\).

Step-by-step solution

Write the Augmented Matrix

For the linear system \(3x + y + 4z = 3\) and \(4x + 3y + z = 1\), the augmented matrix is:\[\begin{bmatrix}3 & 1 & 4 & | & 3 \4 & 3 & 1 & | & 1\end{bmatrix}\]

Reduce Matrix Over \(\mathbb{Z}_5\)

- The augmented matrix under \(\mathbb{Z}_5\) is:\[\begin{bmatrix}3 & 1 & 4 & | & 3 \4 & 3 & 1 & | & 1\end{bmatrix}\]- Divide the first row by 3 (inverse of 3 in \(\mathbb{Z}_5\) is 2):\[\begin{bmatrix}1 & 2 & 3 & | & 1 \4 & 3 & 1 & | & 1\end{bmatrix}\]- Subtract 4 times the first row from the second row:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \0 & 0 & 2 & | & 2\end{bmatrix}\]- Divide second row by 2:\[\begin{bmatrix}1 & 2 & 3 & | & 1 \0 & 0 & 1 & | & 1\end{bmatrix}\]- Subtract 3 times the second row from the first row:\[\begin{bmatrix}1 & 2 & 0 & | & 3 \0 & 0 & 1 & | & 1\end{bmatrix}\]

Solve Reduced System Over \(\mathbb{Z}_5\)

The reduced matrix corresponds to:- \(x + 2y = 3\)- \(z = 1\)From the first equation, solve for \(x\):- \(x = 3 - 2y\)General solution in \(\mathbb{Z}_5\):\((x, y, z) = (3 - 2y, y, 1)\)

Reduce Matrix Over \(\mathbb{Z}_7\)

- The augmented matrix under \(\mathbb{Z}_7\) is:\[\begin{bmatrix}3 & 1 & 4 & | & 3 \4 & 3 & 1 & | & 1\end{bmatrix}\]- Divide the first row by 3 (inverse of 3 in \(\mathbb{Z}_7\) is 5):\[\begin{bmatrix}1 & 5 & 6 & | & 1 \4 & 3 & 1 & | & 1\end{bmatrix}\]- Subtract 4 times the first row from the second row:\[\begin{bmatrix}1 & 5 & 6 & | & 1 \0 & 4 & 1 & | & 3\end{bmatrix}\]- Divide second row by 4 (inverse of 4 in \(\mathbb{Z}_7\) is 2):\[\begin{bmatrix}1 & 5 & 6 & | & 1 \0 & 1 & 2 & | & 6\end{bmatrix}\]- Subtract 5 times the second row from the first row:\[\begin{bmatrix}1 & 0 & 4 & | & 3 \0 & 1 & 2 & | & 6\end{bmatrix}\]

Solve Reduced System Over \(\mathbb{Z}_7\)

The reduced matrix corresponds to:- \(x + 4z = 3\)- \(y + 2z = 6\)From the first equation, solve for \(x\):- \(x = 3 - 4z\)From the second equation, solve for \(y\):- \(y = 6 - 2z\)General solution in \(\mathbb{Z}_7\):\((x, y, z) = (3 - 4z, 6 - 2z, z)\)

Key concepts

Reduced Row-Echelon Form

Reduced Row-Echelon Form (RREF) is an essential concept in linear algebra that gives a systematic way to solve linear systems. The goal is to transform a given matrix into a simplified format where it can easily be read to determine the solution of the system. This method involves performing row operations to simplify the matrix:

• Switching the positions of two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting multiples of rows from each other.

A matrix is in RREF when:

• The first nonzero number in each row, called a leading entry, is 1.
• Each leading 1 is the only non-zero number in its column.
• The leading 1 in any row is to the right of the leading 1 in the row above it.
• Rows with all zero elements, if any, are at the bottom of the matrix.

This form is particularly useful because it makes it very simple to observe the solutions to a system, allowing you to clearly see dependencies between the variables. In practice, reaching the RREF involves lots of careful arithmetic often requiring detailed attention to arithmetic operations, especially over different fields.

Finite Field Arithmetic

Finite Field Arithmetic refers to performing calculations within a set that contains a finite number of elements, often denoted as \(\mathbb{Z}_p\), where \(p\) is a prime number. The arithmetic in this field is done modulo \(p\), meaning that numbers wrap around when they reach \(p\).

In linear algebra, it's crucial to understand finite fields when solving systems over different fields, such as \(\mathbb{Z}_5\) or \(\mathbb{Z}_7\). Each field has its unique set of rules for addition, subtraction, multiplication, and inversation, all conducted under mod \(p\). Here’s how it works:

• Addition: Add two numbers and find the remainder when divided by \(p\).
• Subtraction: Subtract the numbers, adjust if negative by adding \(p\), then take mod \(p\).
• Multiplication: Multiply the numbers and take mod \(p\).
• Inversion: Find a number which, when multiplied by the given number, yields 1. This is crucial for operations like dividing a row in a matrix.

Working in a finite field requires stark attention to these rules, as they differ from standard arithmetic primarily due to the modulus operation that clamps all calculations within the field’s limits.

Linear Systems

Linear systems are collections of linear equations involving the same set of variables. In typical terms, you might see these written as:

• \(a_1x + b_1y + ... = c_1\)
• \(a_2x + b_2y + ... = c_2\)

The goal is to find values for the variables \(x, y, ... \) that satisfy all the equations simultaneously.

To solve these systems, mathematicians and scientists use techniques such as Gaussian elimination and transforming the system into a matrix format. The matrix approach often involves converting the system into an augmented matrix where columns represent the coefficients of the variables and rows represent the equations.

By performing row operations, we can analyze complex linear systems succinctly and identify conditions for solutions or inconsistencies within the system. For instance, finding the reduced row-echelon form gives us a clearer path to understand how variables depend on one another or if multiple solutions exist especially over different finite fields.

Modular Arithmetic

Modular Arithmetic is a central concept in number theory and computer science. It involves arithmetic operations where numbers wrap around after reaching a certain value, known as the modulus.

Operations in modular arithmetic include:

• Addition and subtraction: Like regular arithmetic but results are reduced by the modulus.
• Multiplication: Works the same as usual, but output is reduced by the modulus.
• Division: More complicated, requires finding multiplicative inverses.

For example, if working in \(\mathbb{Z}_5\), numbers cycle back to zero after four, leading to a unique set of results when performing regular arithmetic operations using mod 5. If we add 3 and 4, we get 7, which modulo 5 gives 2.

Modular arithmetic is particularly useful in fields such as cryptography, coding theory, and solving algebraic equations over finite fields. When solving linear systems over fields such as \(\mathbb{Z}_5\) or \(\mathbb{Z}_7\), understanding how to apply modular arithmetic correctly is key to finding correct and valid solutions. This ensures all calculations maintain consistency within the finite nature of the field being used.