Problem 15 - Advanced Engineering Mathematics

Question

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=x, \quad f_{2}(x)=x^{2}, \quad f_{3}(x)=4 x-3 x^{2} $$

Step-by-Step Solution

Answer

The set of functions is linearly independent.

Step 1: Setting Up the Problem

To determine if the set of functions \(f_1(x) = x\), \(f_2(x) = x^2\), and \(f_3(x) = 4x - 3x^2\) is linearly independent, consider the equation \(c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0\) for all \(x\), where \(c_1, c_2, \) and \(c_3\) are constants. Our goal is to find if there is a non-trivial solution (i.e., not all \(c_i = 0\)) for this equation.

Step 2: Formulate the Equation

Substitute the given functions into the equation: \(c_1 x + c_2 x^2 + c_3 (4x - 3x^2) = 0\). Expand and combine like terms: \(c_1 x + c_2 x^2 + 4c_3 x - 3c_3 x^2 = 0\), which simplifies to \((c_1 + 4c_3) x + (c_2 - 3c_3) x^2 = 0\).

Step 3: Simplify and Identify Coefficients

For the above equation to be true for all \(x\), the coefficients of each power of \(x\) must be zero. This gives us a system of equations: \(c_1 + 4c_3 = 0\) and \(c_2 - 3c_3 = 0\).

Step 4: Solve the System of Equations

Solve the system of equations: \(c_1 + 4c_3 = 0\) implies \(c_1 = -4c_3\) and \(c_2 - 3c_3 = 0\) implies \(c_2 = 3c_3\). Substitute \(c_1 = -4c_3\) and \(c_2 = 3c_3\) back into our problem, we observe that if \(c_3 = 0\), then \(c_1\) and \(c_2\) must also be zero. Thus, the only solution is the trivial solution \(c_1 = c_2 = c_3 = 0\).

Step 5: Conclusion

Since the only solution to the system is the trivial solution, the functions \(f_1(x) = x\), \(f_2(x) = x^2\), and \(f_3(x) = 4x - 3x^2\) are linearly independent on the interval \((-fty, fty)\).

Key Concepts

System of Equations

In mathematics, a system of equations is a set of equations with multiple variables. The primary goal is to find the values for these variables that satisfy all the equations simultaneously. Let's break it down using our specific exercise:

We have three functions: \(f_1(x) = x\), \(f_2(x) = x^2\), and \(f_3(x) = 4x - 3x^2\). To determine if they are linearly independent, we consider a linear combination of these functions set to zero:

• \(c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0\)

Where \(c_1, c_2, \) and \(c_3\) are constants. This translates to the equation: \((c_1 + 4c_3)x + (c_2 - 3c_3)x^2 = 0\) for all \(x\).

This setup imposes a system of equations, representing each separate power of \(x\):

• Equation 1: \(c_1 + 4c_3 = 0\)
• Equation 2: \(c_2 - 3c_3 = 0\)

The system of equations helps us explore whether a unique solution exists, which affects the linear independence of the functions.

Function Analysis

When analyzing functions, you delve into their behavior, expressions, and inter-relations. Here, function analysis is critical as we examine three distinct functions: \(f_1(x) = x\), \(f_2(x) = x^2\), and \(f_3(x) = 4x - 3x^2\). In considering whether these functions are linearly dependent or independent, we analyze their structure and interaction.

Firstly, each function represents a different polynomial degree:

• \(f_1(x)\) is a first-degree polynomial.
• \(f_2(x)\) is a second-degree polynomial.
• \(f_3(x)\) is a combination of first and second-degree terms.

The analysis involves substituting them into a linear combination:\(c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0\), and exploring how these functions overlap or cancel out. Through function analysis, we discover that the interaction among these functions can yield a system of equations, crucial in determining linear independence.

Trivial Solution

A trivial solution in the context of linear equations is the simplest form, where all constants equal zero. For our problem, it involves checking whether the only solution is \((c_1, c_2, c_3) = (0, 0, 0)\). This is vital because:

If the only solution to our system of equations is this trivial one, then the functions are linearly independent.

In our specific exercise, after formulating and simplifying the system of equations derived from the functions:

• \(c_1 + 4c_3 = 0\)
• \(c_2 - 3c_3 = 0\)

By examining these equations, setting \(c_3 = 0\), leads straight to \(c_1 = 0\) and \(c_2 = 0\). The solution cannot be any other than the trivial one. Recognizing a trivial solution is an indicator of linear independence and helps affirm that these functions are distinct without overlap on the interval \((-fty,fty)\).

Non-trivial Solution

A non-trivial solution refers to solutions where not all constants are zero. Its presence typically indicates that the functions involved are linearly dependent. However, identifying or discounting non-trivial solutions is crucial to understand linear relationships.

In our exercise, the aim was to find any non-trivial solution to the system:

• \(c_1 + 4c_3 = 0\)
• \(c_2 - 3c_3 = 0\)

Substituting values like \(c_3\) to 0, reveals every path leads to \(c_1 = 0\) and \(c_2 = 0\). This path of exhaustion confirms our equation only supports the trivial solution. Therefore, no such non-trivial solution exists in our scenario, reinforcing the conclusion of linear independence. Understanding this concept ensures clarity about the distinct behavior of the given functions over the entire real number line.