Use a computer algebra system to solve the system w+x+y+z=23 w+2 x+3 y+4 z=19 w

Use a computer algebra system to solve the system ...

Key Concepts

Computer Algebra Systems (CAS)

A computer algebra system is software designed to facilitate symbolic mathematics. CAS tools can automatically handle operations like Gaussian elimination in modular arithmetic, simplifying the process of solving systems of linear equations, especially when dealing with complications introduced by composite moduli.

Composite Modulus and Finite Rings

When the modulus, such as 91, is composite (91 = 7 × 13), the set of integers modulo that number forms a ring rather than a field. This means some nonzero elements may not have an inverse, impacting the methods used in solving linear systems and sometimes necessitating alternative techniques or the use of the Chinese Remainder Theorem.

Gaussian Elimination in Modular Systems

Gaussian elimination is a method for solving systems of linear equations by applying row operations to reduce the matrix to an easier form (such as row-echelon form). When performed modulo a composite number, extra care must be taken since not every nonzero element possesses a multiplicative inverse, which can complicate the elimination process.

Matrix Representation in Modular Arithmetic

Representing a system of equations in matrix form, where the coefficient matrix, variable vector, and constant vector are all taken modulo a number, streamlines the process of visualizing and solving the set of congruences. This approach allows the use of linear algebra techniques adapted for modular arithmetic.

Systems of Linear Equations in Modular Settings

A system of linear equations in modular arithmetic involves finding solutions for variables such that each equation holds true modulo a given integer. In such systems, coefficients, variables, and constants are all considered modulo the modulus, and standard operations are adapted accordingly.

Modular Arithmetic

This concept deals with integers considered under a modulus—in this case 91—where numbers “wrap around” after reaching the modulus value. In modular arithmetic, two numbers are equivalent if their difference is a multiple of the modulus, and all arithmetic operations (addition, subtraction, multiplication) are performed with the modulus in mind.

Transcript

00:03 Let's make a matrix, 4, negative 2, 10, negative 3, negative 1, 2, negative 5, 2, and 2, and 15.

00:23 Let's do ghost jordan, actually, row 2, is there a smallest entry, so why don't i just divide out by negative 2? so i'll end up with 1, 1 half, negative 1, negative 1, negative 7 over 2, negative 1.

00:43 And now let's multiply it by 4, subtract it from row 1.

00:47 So we'll get a 0, negative 5, 5, 15, 25.

00:54 Leave, let's see.

00:59 I'll multiply it by 10, subtract it from row 3.

01:03 0, negative 5, 5.

01:09 Okay, notice that's just like the top row.

01:12 So on the next step, what i'm going to do is choose the top row.

01:15 And then i will divide it out by well first let's just get rid of the bottom row we're ready now it's going to happen because i can just subtract it in row one from row three and then we'll just have zero zero zero zero zero zero and then i can divide out by five or negative five end up with zero one negative one negative three negative five okay negative seven over two negative one well actually let's divide it by 2 and subtracted from our central rule.

01:53 So we'll end up with 0 -1, negative 1, negative 3, negative 5.

02:01 Remember we divide by 2, so we'll have 1 -0, divide them by 2 and subtract it from the bottom.

02:11 We'll have negative 1 1 .5, and then we'll plus 3 over 2 to that...

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