Problem 42 - Differential Equations and Linear Algebra

Question

Show that the functions $$ f_{1}(x)=\left\\{\begin{aligned} x-1, & \text { if } x \geq 1 \\ 2(x-1), & \text { if } x<1 \end{aligned}\right. $$ \(f_{2}(x)=2 x, f_{3}(x)=3\) form a linearly independent set on \((-\infty, \infty) .\) Determine all intervals on which \(\left\\{f_{1}, f_{2}, f_{3}\right\\}\) is linearly dependent.

Step-by-Step Solution

Answer

The functions \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\) are linearly independent on the intervals \((1, \infty)\) and \((-\infty, 1)\).

Step 1: Write the linear combination of the functions

Let's consider a linear combination of \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\): $$a f_1(x) + b f_2(x) + c f_3(x) = 0$$ where \(a, b, c\) are constants.

Step 2: Check if the linear combination is zero for \(x \geq 1\)

For \(x \geq 1\), \(f_1(x) = x - 1\), so our linear combination becomes: $$a(x - 1) + 2bx + 3c = 0$$

Step 3: Check if the linear combination is zero for \(x < 1\)

For \(x < 1\), \(f_1(x) = 2(x - 1)\), so our linear combination becomes: $$2a(x - 1) + 2bx + 3c = 0$$

Step 4: Determine if the functions are linearly independent

Let us now analyze the LHS of these equations to determine if \(a = 0, b = 0\), and \(c = 0\) are the only solutions. If we set all coefficients to zero in both the equations, we get: $$a - 2b = 0$$ $$2a - 2b = 0$$ $$c = 0$$ Note that the second equation is equivalent to the first equation. Therefore, we only have one independent equation relating \(a\) and \(b\). This implies the possibility of having non-trivial solutions \((a, b, c)\), and thus the functions are linearly dependent on some interval.

Step 5: Determine the intervals of linear dependence

We now need to determine the intervals where these functions are linearly dependent. First, focus on the linear combination for \(x \geq 1\) which is \(a(x - 1) + 2bx + 3c = 0\). We can rewrite this equation as $$a(x-1)+2b(x-1)+3c=2b$$ which means, \((a+2b-2b)(x-1)=2b -3c\) Since this must hold for all \(x > 1\), we conclude that \(a+2b-2b = 0\) and \(2b -3c = 0\). The first equality yields a trivial solution. Now, focus on the linear combination for \(x < 1\) which is \(2a(x - 1) + 2bx + 3c = 0\). As before, we can rewrite this equation as $$2a(x-1)+2b(x-1)+3c=-2b$$ which means, \((2a+2b-2b)(x-1)=-2b+3c\) Similarly, since this must hold for all \(x < 1\), we conclude that \(2a+2b-2b = 0\) and \(-2b+3c = 0\). The first equality yields another trivial solution. Therefore, the functions \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\) are linearly independent on the intervals \((1, \infty)\) and \((-\infty, 1)\).

Key Concepts

Piecewise Functions

Piecewise functions are a fascinating type of function where different expressions are used for different parts of the domain. In the original exercise, the function \( f_1(x) \) is defined as a piecewise function with two different expressions depending on whether \( x \) is greater than or equal to 1 or less than 1.
These functions can be quite useful because they allow different types of behavior within the same function. This is particularly helpful in mathematical modeling, where systems often behave differently under different conditions.
For instance, the piecewise function \( f_1(x) \) helps accommodate changes in its expression at the point where \( x = 1 \). If you plot \( f_1(x) \), you'll notice that it represents two distinct linear functions each covering a specific half of the domain. The use of piecewise functions like \( f_1(x) \) provides flexibility and precision in defining functions that are otherwise not achievable with single expressions.

Linear Combinations

A linear combination involves creating a new function by summing scalar multiples of several others. The main purpose is to determine if these functions can be expressed in terms of one another using constants.
In the context of this exercise, we expressed the linear combination of \( f_1(x) \), \( f_2(x) \), and \( f_3(x) \) as \( a f_1(x) + b f_2(x) + c f_3(x) = 0 \). Here, \( a \), \( b \), and \( c \) are the scalar coefficients.
This expression is used to explore whether the functions are linearly independent or dependent. If the only solution to the equation \( a f_1(x) + b f_2(x) + c f_3(x) = 0 \) with varying \( x \) values is when all coefficients are zero, then these functions are linearly independent. Linear combinations are thus powerful tools in determining the relationship and dependence between functions.

Intervals of Dependence

Intervals of dependence refer to specific ranges in the domain where the functions lose their independence due to linear dependencies among them. Assessing these intervals is crucial for analyzing the behavior of functions in various segments of the domain.
For this problem, we examined two primary intervals based on the piecewise nature of \( f_1(x) \): \( x \geq 1 \) and \( x < 1 \). Within each interval, specific linear combinations of the functions are scrutinized to check for zero-out conditions.
Through these steps, we determined if there were any intervals where the functions became linearly dependent—meaning they could be expressed as a combination of each other. The results showed distinct behaviors based on these domain intervals, leading to an understanding of where functions maintain independence.

Linearly Independent Sets

A set of functions is deemed linearly independent if no function in the set can be written as a linear combination of the others. This concept is vital in many areas of mathematics, including dimensionality and system solvability.
In the set \( \{f_1, f_2, f_3\} \), determining if they are linearly independent involves checking if the coefficients \( a = 0, b = 0, \) and \( c = 0 \) are the only solutions to their linear combinations across all possible \( x \) values.
For this exercise, despite intricacies in intervals \((x < 1)\) and \((x \geq 1)\), an analysis of the coefficients showed the set retains independence in both regions of \( x \). Thus, the functions \( f_1, f_2, \) and \( f_3 \) are indeed a linearly independent set across the entire real number line. Understanding linear independence helps classify functions, ensuring they provide unique, non-redundant contributions.