Problem 13 - Differential Equations and Linear Algebra

Question

Write the system of equations with the given coefficient matrix and right-hand side vector. $$A=\left[\begin{array}{rrr}2 & 1 & 3 \\\4 & -1 & 2 \\\7 & 6 & 3\end{array}\right], \mathbf{b}=\left[\begin{array}{r}3 \\\1 \\\\-5\end{array}\right]$$

Step-by-Step Solution

Answer

The given system of equations corresponding to the provided coefficient matrix A and right-hand side vector b is: \[ 2x + y + 3z = 3 \] \[ 4x - y + 2z = 1 \] \[ 7x + 6y + 3z = -5 \]

Step 1: Identify the order of the matrix

First, we need to identify the order of the matrix A. In this case, A is a 3x3 matrix because it has 3 rows and 3 columns.

Step 2: Identify coefficients from the matrix

Next, we will extract the coefficients for the unknowns (x, y, z) from the matrix A. We can recognize the coefficients by identifying the elements of A belonging to each equation: - Equation 1 coefficients: 2, 1, and 3 (from the first row of A) - Equation 2 coefficients: 4, -1, and 2 (from the second row of A) - Equation 3 coefficients: 7, 6, and 3 (from the third row of A)

Step 3: Identify the constants from the right-hand side vector

Now, we need to extract the constants from the right-hand side vector b. The constants are in the following order: - Equation 1 constant: 3 - Equation 2 constant: 1 - Equation 3 constant: -5

Step 4: Write the system of equations using the identified coefficients and constants

Finally, we can use the identified coefficients and constants to write the system of equations: \[ 2x + 1y + 3z = 3 \] \[ 4x - 1y + 2z = 1 \] \[ 7x + 6y + 3z = -5 \] So, the given system of equations corresponding to the provided coefficient matrix A and right-hand side vector b is: \[ 2x + y + 3z = 3 \] \[ 4x - y + 2z = 1 \] \[ 7x + 6y + 3z = -5 \]

Key Concepts

Matrix Operations

Matrix operations are fundamental in the study and application of linear algebra. They allow us to solve systems of linear equations effectively by using various techniques. Matrices can be added, subtracted, and multiplied, and each of these operations has specific rules. For example:

• Addition: Two matrices can be added if and only if they are of the same dimension. The addition is done element-wise.
• Subtraction: Similarly, subtraction between two matrices is carried out element-by-element, provided the matrices are of the same size.
• Multiplication: Matrix multiplication is more complex. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. The result is a new matrix where each element is calculated as the sum of products of corresponding elements from the rows of the first matrix and the columns of the second matrix.

These operations are often used in conjunction with techniques like Gaussian elimination or matrix inversion to find solutions to systems of equations. By using matrix operations, we can simplify complex systems into a manageable form, making it easier to understand the relationships between multiple variables.

Coefficient Matrix

A coefficient matrix is an integral part of solving systems of linear equations. It represents all the coefficients of the variables in a system and allows us to visualize and manipulate these systems efficiently. Consider the matrix:\[A=\begin{bmatrix}2 & 1 & 3 \4 & -1 & 2 \7 & 6 & 3\end{bmatrix}\]In this structure, each row of the matrix corresponds to an equation from the system, and each column corresponds to one of the variables (e.g., \(x\), \(y\), \(z\)). This organization allows various techniques to be applied more seamlessly:

• Using the coefficient matrix, we can apply methods such as row reduction, which simplifies the system to find solutions.
• The matrix can be combined with the right-hand side vector \(\mathbf{b}\) to create an augmented matrix, which is a key step in solving systems.

Overall, the coefficient matrix offers a structured and efficient way to handle and analyze complex systems of linear equations, helping us understand their solutions and properties.

Vector Spaces

Vector spaces are fundamental constructs in linear algebra, providing a comprehensive framework for analyzing systems of linear equations, transformations, and more. A vector space is a collection of vectors that can be added together and multiplied by scalars to produce new vectors.

• Vectors: These are elements within a vector space that can be scaled and added while remaining within the space. For example, in a three-dimensional space, a vector might look like \([x, y, z]\).
• Dimensions: This refers to the number of vectors in the basis for the space. The basis is the set of linearly independent vectors that span the space. Each dimension corresponds to a vector in the basis.
• Properties: Vector spaces must adhere to specific properties, like commutativity of vector addition and compatibility of scalar multiplication, among others.

In relation to systems of equations, understanding vector spaces allows us to comprehend solutions and their uniqueness. For instance, whether a particular vector is within the span of a set of other vectors (i.e., its linear independence or dependence) helps determine the solutions' existence and consistency. Essentially, vector spaces provide a structured environment that supports operations on systems and solutions in linear algebra.