Problem 40 - Calculus for Biology and Medicine

Question

Write each system in matrix form. $$ \begin{array}{r} 2 x_{2}-x_{1}=x_{3} \\ 4 x_{1}+x_{3}=7 x_{2} \\ x_{2}-x_{1}=x_{3} \end{array} $$

Step-by-Step Solution

Answer

The system in matrix form is \( \begin{bmatrix} -1 & 2 & -1 \\ 4 & -7 & 1 \\ -1 & 1 & -1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \).

Step 1: Identify the Variables

In the system of equations, identify the variables involved: \(x_1\), \(x_2\), and \(x_3\).

Step 2: Write Each Equation in Standard Form

Each equation should be rewritten in the form \(Ax_1 + Bx_2 + Cx_3 = D\). The given equations are:1. \(-x_1 + 2x_2 - x_3 = 0\)2. \(4x_1 - 7x_2 + x_3 = 0\)3. \(-x_1 + x_2 - x_3 = 0\)

Step 3: Extract the Coefficient Matrix

Extract the coefficients from each equation to form the coefficient matrix. For the matrix, each row corresponds to one equation: \[\begin{bmatrix}-1 & 2 & -1 \4 & -7 & 1 \-1 & 1 & -1\end{bmatrix}\]

Step 4: Form the Vector for Variables and Constants

Now, form the two separate vectors. One for the variables \([x_1, x_2, x_3]^T\) and another for the constants on the right side, which is \([0, 0, 0]^T\), since each equation equals 0.

Step 5: Combine into Matrix Form

Combine the coefficient matrix, the variable vector, and the constant vector into the matrix equation form: \[\begin{bmatrix}-1 & 2 & -1 \4 & -7 & 1 \-1 & 1 & -1\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}=\begin{bmatrix}0 \0 \0\end{bmatrix}\]

Key Concepts

System of Equations

A system of equations is merely a set of equations with multiple variables. In the exercise provided, we have three equations with three variables: \(x_1\), \(x_2\), and \(x_3\). Each equation provides a unique constraint involving these variables which must be satisfied simultaneously.

To solve such systems, one must find a common solution, that is values for \(x_1\), \(x_2\), and \(x_3\) that satisfy all the equations at once. This is typically done using methods from linear algebra, such as substitution, elimination, or matrix operations.

Understanding a system of equations is crucial for solving real-world problems, as they can represent conditions in engineering, economics, and science among many other fields. When dealing with larger systems, matrix form becomes especially useful, allowing for more structured computation.

Coefficient Matrix

The coefficient matrix is a key component when expressing a system of equations in matrix form. It consists only of the coefficients of the variables from each equation. This makes it much easier to apply linear algebra techniques.

Here’s how you construct it:

• The first row in the matrix corresponds to the coefficients in the first equation.
• The second row corresponds to the second equation, and so on.

For example, our system gives rise to the coefficient matrix:\[\begin{bmatrix} -1 & 2 & -1 \4 & -7 & 1 \-1 & 1 & -1 \end{bmatrix}\]The coefficient matrix provides a compact way to represent the system’s structure, separating variable interactions from constant terms. It is fundamental for applying matrix algebra techniques.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear transformations. It provides powerful tools for analyzing systems of linear equations like the one in our exercise.

Some fundamental concepts of linear algebra include:

• Vectors: These are objects that have both direction and magnitude. In our exercise, the variable vector \([x_1, x_2, x_3]^T\) and constant vector \([0, 0, 0]^T\) are examples.
• Matrices: These are rectangular arrays of numbers used to represent linear mappings. The coefficient matrix is an example of a matrix.

Using linear algebra, we can perform operations on matrices to find solutions to systems by various methods such as Gaussian elimination, matrix inversion, or determinants. This provides a systematic method to solve complex problems far beyond simple manual calculations, especially for high-dimensional data. Linear algebra remains essential in multiple fields including computer graphics, engineering, and data science.