Problem 11 - Precalculus: Mathematics for Calculus

Question

Write the augmented matrix for the system of linear equations. $$\left\\{\begin{array}{c} 3 x+y-z=2 \\ 2 x-y \quad=1 \\ x-z=3 \end{array}\right.$$

Step-by-Step Solution

Answer

The augmented matrix is \(\begin{bmatrix} 3 & 1 & -1 & | & 2 \\ 2 & -1 & 0 & | & 1 \\ 1 & 0 & -1 & | & 3 \end{bmatrix}\).

Step 1: Identify Coefficients

To write the augmented matrix for a system of linear equations, first identify the coefficients of each variable (and the constant on the right) in each equation. For the given system, the equations are: 1. \(3x + y - z = 2\)2. \(2x - y = 1\)3. \(x - z = 3\).The coefficients for \(x\), \(y\), and \(z\) for each equation, respectively, are: - (3, 1, -1, 2)- (2, -1, 0, 1)- (1, 0, -1, 3).

Step 2: Form the Augmented Matrix

Using the coefficients identified in the previous step, form an augmented matrix. This matrix includes the coefficients of each variable for each equation as separate rows, as well as the constants on the right of the equations as the last column.The augmented matrix is:\[\begin{bmatrix}3 & 1 & -1 & | & 2 \2 & -1 & 0 & | & 1 \1 & 0 & -1 & | & 3\end{bmatrix}\]

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. In our given example, we have three equations with variables \(x\), \(y\), and \(z\). Each equation represents a line or plane in a multidimensional space, and the solution to the system is the set of values for \(x\), \(y\), and \(z\) that satisfies all the equations simultaneously.
Solving a system of linear equations can help in finding points of intersection of these lines or planes. Systems can be consistent, meaning they have at least one solution, or inconsistent, with no solutions at all.

• If there is exactly one solution, the system is determined as independent.
• If there are infinitely many solutions, the equations are dependent, meaning they represent the same geometrical object.
• When there is no solution, the lines or planes are parallel.

Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are a key tool in linear algebra, used to perform various operations such as addition, subtraction, and multiplication. In our example, an augmented matrix is utilized to compactly represent a system of linear equations. The augmented matrix for the system given is:\[\begin{bmatrix} 3 & 1 & -1 & | & 2 \2 & -1 & 0 & | & 1 \1 & 0 & -1 & | & 3 \end{bmatrix}.\]Here, the vertical bar separates the coefficients on the left from the constants on the right.
This structure facilitates systematic manipulation and simplification of the system's equations. Often, matrices are used to apply operations like row reduction, which helps in solving systems efficiently.
Matrix operations are foundational for computer graphics, scientific simulations, and solving linear equation systems.

Coefficients

Coefficients are the numerical factors that multiply the variables in an equation. Identifying these coefficients is crucial for forming matrices.
In our system of equations:

• The coefficient of \(x\) in the first equation is 3.
• The coefficient of \(y\) is 1.
• The coefficient of \(z\) is -1.
• And the constant term on the right is 2.

Coefficients express the proportional relationship between the variables and help in forming equations into a matrix.
Understanding how to identify and organize these coefficients into an augmented matrix allows easier handling of complex systems. The matrix keeps the system's information structured, making operations like Gaussian elimination straightforward.

Linear Algebra

Linear algebra is a branch of mathematics concerned with vectors, vector spaces, and linear equations. It provides powerful tools for analyzing and solving systems of linear equations. By using matrices and determinants, linear algebra simplifies complex computations and enables efficient problem-solving.
It introduces fundamental concepts such as:

• Vectors and operations involving them.
• Matrix algebra, illustrating how matrices can be manipulated.
• Determinants and eigenvalues, which offer insights into matrix properties and transformations.

Linear algebra is instrumental in diverse fields from computer science to engineering and physics.
Understanding the basics of systems of equations and matrices gives a strong foundation for exploring more advanced topics within this field, including transformations and vector spaces.