Problem 19 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The set of functions is linearly dependent.

Step 1: Understanding the concept of linear dependence and independence

A set of functions \( f_1, f_2, \ldots, f_n \) is said to be linearly dependent if there exist constants \( c_1, c_2, \ldots, c_n \), not all zero, such that \( c_1f_1(x) + c_2f_2(x) + \cdots + c_nf_n(x) = 0 \) for all values of \( x \). If no such constants exist, then the functions are linearly independent.

Step 2: Set up the linear combination equation

For the functions \( f_1(x) = x \), \( f_2(x) = x - 1 \), and \( f_3(x) = x + 3 \), set up the equation \( c_1f_1(x) + c_2f_2(x) + c_3f_3(x) = 0 \).This simplifies to:\[ c_1x + c_2(x - 1) + c_3(x + 3) = 0 \].

Step 3: Simplify the equation

Expand and simplify the expression:\[ c_1x + c_2x - c_2 + c_3x + 3c_3 = 0 \].Combine like terms to produce:\[ (c_1 + c_2 + c_3)x + (-c_2 + 3c_3) = 0 \].

Step 4: Solve for constants to determine linear dependence

To satisfy the equation for all \( x \), we set the coefficients of \( x \) and the constant term to zero:1. \( c_1 + c_2 + c_3 = 0 \)2. \( -c_2 + 3c_3 = 0 \)From equation 2, solve for \( c_2 \):\[ c_2 = 3c_3 \].

Step 5: Substitute and find the values for constants

Substitute \( c_2 = 3c_3 \) into equation 1:\[ c_1 + 3c_3 + c_3 = 0 \]\[ c_1 + 4c_3 = 0 \].Solve for \( c_1 \):\[ c_1 = -4c_3 \].Choosing \( c_3 = 1 \), we find \( c_1 = -4 \) and \( c_2 = 3 \).

Step 6: Verify the solution

Substitute the constants back into the original equation:\[ -4x + 3(x - 1) + (x + 3) = 0 \].This simplifies to:\[ -4x + 3x - 3 + x + 3 = 0 \]\[ 0 = 0 \].The numbers check out, confirming the functions are linearly dependent.

Key Concepts

Linearly Independent

When we talk about sets of functions, linear independence is a key concept. Imagine you have several functions, like in our example: \( f_1(x)=x \), \( f_2(x)=x-1 \), and \( f_3(x)=x+3 \). These functions are said to be linearly independent if no non-trivial combination of them (meaning some of the constants are not zero) can add up to the zero function when combined. If this happens, it means you can't express one function as a combination of the others.
If the functions are linearly independent, all coefficients in front of the functions must be zero in the linear combination because there's no way to make a zero function otherwise.
This concept is very important because it tells us about the uniqueness of representation using these functions.

Linear Combination

A linear combination is a mathematical way to blend several functions or vectors. Picture it like creating a recipe with your functions as ingredients. The combination involves constants - like spices you add to taste - that multiply each function to create a desired result. Formally, if you have functions \( f_1(x), f_2(x), \ldots, f_n(x) \), a linear combination is something like \( c_1f_1(x) + c_2f_2(x) + \ldots + c_nf_n(x) \).
The challenge in our exercise was to determine if these could create the zero function without all constants being zero, i.e., can a non-trivial mix result in zero? In our solution, with constants \( c_1 = -4, c_2 = 3, c_3 = 1 \), the answer was yes, indicating dependence. This verification shows us how mixing functions with specific constants can sometimes entirely cancel them out, demonstrating dependency.

Concept of Dependence in Functions

Linear dependence in functions occurs when you can express one function as a linear combination of others. This means there are constants, not all zero, making this expression possible. Illustrated in our exercise, the functions \( f_1(x)=x \), \( f_2(x)=x-1 \), and \( f_3(x)=x+3 \) were found to be dependent.
Here's why dependency is useful:

• It's a signal that functions aren't adding new dimensions of information but might be redundant or representable by others.
• This understanding refines models, ensuring only unique, independent data is considered for plotting or calculations.

In the solved exercise, by verifying that certain constant values satisfy the dependence condition, it proved that not all functions shared fresh information or dimensions.