Problem 20 - Advanced Engineering Mathematics

Question

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=2+x, \quad f_{2}(x)=2+|x| $$

Step-by-Step Solution

Answer

The functions are linearly independent on \((\nolinebreak-\infty, \nolinebreak\infty)\).

Step 1: Understand the Problem

To determine if the functions \( f_1(x) = 2 + x \) and \( f_2(x) = 2 + |x| \) are linearly dependent on \((-olinebreak\infty, olinebreak\infty)\), we need to find if there exists non-zero constants \( c_1 \) and \( c_2 \) such that \( c_1 f_1(x) + c_2 f_2(x) = 0 \) for all \( x \).

Step 2: Set Up the Linear Combination

Consider the linear combination: \( c_1(2 + x) + c_2(2 + |x|) = 0 \). Expand this equation to get \( (2c_1 + 2c_2) + c_1x + c_2|x| = 0 \) for all \( x \).

Step 3: Analyze the Absolute Value Function

Recognize that \( |x| \) changes behavior at \( x = 0 \). For \( x \geq 0 \), \( |x| = x \); for \( x < 0 \), \( |x| = -x \). We should check both cases.

Step 4: Case 1 - Assume \( x \geq 0 \)

Here, \( |x| = x \). Substitute into the equation to get: \( (2c_1 + 2c_2) + (c_1 + c_2)x = 0 \). Since the equation must be true for all \( x \geq 0 \), both coefficients must be zero.

Step 5: Simplify Case 1

We have the system of equations: \( 2c_1 + 2c_2 = 0 \) and \( c_1 + c_2 = 0 \). Solving, we find \( c_1 = 0 \) and \( c_2 = 0 \).

Step 6: Case 2 - Assume \( x < 0 \)

Here, \( |x| = -x \). Substitute into the equation to get: \( (2c_1 + 2c_2) + (c_1 - c_2)x = 0 \). Again, both coefficients must be zero for all \( x < 0 \).

Step 7: Simplify Case 2

This gives the system: \( 2c_1 + 2c_2 = 0 \) and \( c_1 - c_2 = 0 \). Solving these, we also find \( c_1 = 0 \) and \( c_2 = 0 \).

Step 8: Conclusion on Linear Dependence

In both cases, the only solution is \( c_1 = 0 \) and \( c_2 = 0 \). Hence, the functions are linearly independent on \((olinebreak-\infty, olinebreak\infty)\).

Key Concepts

Linear Dependence

When we talk about linear dependence with functions, we refer to the ability to express one function as a linear combination of others using non-zero constants. If two or more functions are linearly dependent, it essentially means one of the functions can be constructed using the others by applying such constants.
In our problem, we want to assess if the functions \(f_1(x) = 2 + x\) and \(f_2(x) = 2 + |x|\) can be described this way over the entire set of real numbers \((-olinebreak\infty, olinebreak\infty)\).
Suppose we can find constants \(c_1\) and \(c_2\), which are not both zero, that satisfy the equation \(c_1 f_1(x) + c_2 f_2(x) = 0\) for all possible values of \(x\). In such a case, the functions would be linearly dependent.
To determine this accurately, we set up a system of equations. This involves plugging the functions into the linear combination and ensuring that the equations hold true for all \(x\). If the only solution to these equations is \(c_1 = 0\) and \(c_2 = 0\), then the functions are linearly independent, meaning you cannot form one using the others by any linear combination.

Absolute Value Function

The absolute value function, represented as \(|x|\), is a crucial concept in determining the linearity of combinations. It behaves differently depending on the value of \(x\), which creates a pivotal change in mathematical expressions.
Here’s how it works:

• If \(x\) is positive or zero (\(x \geq 0\)), the absolute value function \(|x|\) gives you the number itself, i.e., \(|x| = x\).
• If \(x\) is negative (\(x < 0\)), \(|x|\) returns the positive of that number, essentially flipping the sign, i.e., \(|x| = -x\).

When analyzing the linear dependence of \(f_1\) and \(f_2\), we considered both cases of \(x\). This is because the behavior of \(|x|\) directly impacts the equation \((2c_1 + 2c_2) + (c_1 + c_2)x = 0\) in the case of \(x \geq 0\) and \((2c_1 + 2c_2) + (c_1 - c_2)x = 0\) for \(x < 0\).
By examining these separate cases, we ensure completeness in our solution method to determine whether \(c_1\) and \(c_2\) truly result in linear dependence or show independence.

System of Equations

A system of equations involves finding values for variables that satisfy multiple equations simultaneously. In our linear independence problem, we used systems of equations to verify if non-zero constants \(c_1\) and \(c_2\) exist, which would indicate linear dependence.
Here's the breakdown into two cases:

• For \(x \geq 0\), we derived the system:
  • \(2c_1 + 2c_2 = 0\)
  • \(c_1 + c_2 = 0\)
• For \(x < 0\), we derived a slightly different system:
  • \(2c_1 + 2c_2 = 0\)
  • \(c_1 - c_2 = 0\)

By solving these systems, we concluded that \(c_1 = 0\) and \(c_2 = 0\) for both cases, confirming that \(f_1\) and \(f_2\) do not exhibit linear dependence across all \(x\).
This demonstrates how systems of equations provide a structured method for addressing questions of linear combinations and interdependencies in mathematics.