Problem 15 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Linear Dependence

A set of functions \( f_1, f_2, \, \text{and} \, f_3 \) are linearly dependent if there exist constants \( c_1, c_2, \, \text{and} \, c_3 \), not all zero, such that \( c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \) for all \( x \) in the interval.

Step 2: Set Up the Linear Combination

Express the linear combination of the given functions: \[ c_1 x + c_2 x^2 + c_3 (4x - 3x^2) = 0. \] Simplify this expression: \( c_1 x + c_2 x^2 + c_3 (4x - 3x^2) = c_1 x + (c_2 - 3c_3)x^2 + 4c_3 x = 0. \)

Step 3: Combine Like Terms

Combine like terms to get: \[ (c_1 + 4c_3)x + (c_2 - 3c_3)x^2 = 0. \] For this equation to hold for all \( x \), the coefficients of \( x \) and \( x^2 \) must be zero.

Step 4: Solve the System of Equations

Solve the system: 1. \( c_1 + 4c_3 = 0 \) 2. \( c_2 - 3c_3 = 0 \).

Step 5: Analyze the System of Equations

From equation 1, \( c_1 = -4c_3 \). From equation 2, \( c_2 = 3c_3 \). Substitute these into the condition that not all coefficients are zero. If \( c_3 = 0 \), then \( c_1 \) and \( c_2 \) must also be zero, indicating that the only solution is the trivial one where all constants are zero.

Step 6: Conclude Linear Independence

Since the only solution to \( c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \) is \( c_1 = c_2 = c_3 = 0 \), the functions \( f_1, f_2, \) and \( f_3 \) are linearly independent.

Key Concepts

Set of Functions

A set of functions is simply a collection of functions that we are considering in a particular context. For this exercise, we are looking at three functions: \( f_1(x) = x \), \( f_2(x) = x^2 \), and \( f_3(x) = 4x - 3x^2 \). These functions are defined for all real numbers, which means their domain is the interval \(( -\infty, \infty )\).
Understanding whether this set of functions is linearly dependent or independent is crucial for many mathematical applications, such as solving differential equations and performing operations in vector spaces. Our main goal in these problems is often to determine if we can express one function in the set as a combination of the others.
Linear dependence and independence of functions is a fundamental concept when we dive deeper into linear algebra concepts. It helps identify the basis of a function space or understand the relationship between different functions in a system like this one.

Linear Combination

A linear combination refers to an expression made up of a set of functions in which each function is multiplied by a constant, and the results are added together. For the given set \( \{f_1, f_2, f_3\} \), a linear combination would look like \( c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) \). Here, \( c_1, c_2, \) and \( c_3 \) are constants that can be any real numbers.
In this exercise, the goal of finding a linear combination is to see if we can find such constants that not all are zero, making the linear combination equal to zero across the entire domain of the functions. This determines their dependence or independence.
When these constants can only be zero to satisfy the expression, it means the functions are linearly independent because no function in the set can be written as a combination of the others. If there exists even one non-zero constant that can produce a zero linear combination, then the set is linearly dependent.

Systems of Equations

A system of equations is a set of equations with multiple variables that are solved together. In this context, after forming the linear combination, we arrive at equations by equating coefficients of similar terms to zero. This gives us a system to solve for \( c_1, c_2, \) and \( c_3 \).
For this particular example, we obtained:

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

These equations are derived from setting the linear combination \( (c_1 + 4c_3)x + (c_2 - 3c_3)x^2 = 0 \) to zero for all \( x \). Solving this system whether with substitution or another method, we find the relationships between the constants.
The solution here shows that if \( c_3 = 0 \), then \( c_1 = 0 \) and \( c_2 = 0 \) as well, leading us to conclude that the system only has a trivial solution. This further confirms the linear independence of the functions since no non-trivial solution exists.