Problem 17 - Advanced Engineering Mathematics

Question

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=5, \quad f_{2}(x)=\cos ^{2} x, \quad f_{3}(x)=\sin ^{2} x $$

Step-by-Step Solution

Answer

The functions are linearly dependent.

Step 1: Understanding Linear Dependence/Independence

The functions \(f_1(x), f_2(x)\), and \(f_3(x)\) are said to be linearly dependent if there exist constants \(c_1, c_2,\) 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 \(x\). If only the trivial solution \(c_1 = c_2 = c_3 = 0\) satisfies this equation, then the functions are linearly independent.

Step 2: Identifying the Given Functions

Identify the given functions as follows: \(f_1(x)=5\), a constant function; \(f_2(x)=\cos^2 x\); and \(f_3(x)=\sin^2 x\).

Step 3: Using a Trigonometric Identity

Recall the trigonometric identity for any angle \(x\): \(\cos^2 x + \sin^2 x = 1\).

Step 4: Set Up the Linear Combination

Consider the linear combination \(c_1(5) + c_2(\cos^2 x) + c_3(\sin^2 x) = 0\). Substitute \(\cos^2 x + \sin^2 x = 1\) into the equation.

Step 5: Solve the Equation for Constants

Substitute the identity into the linear combination: \(c_1(5) + c_2(\cos^2 x + \sin^2 x) = c_1(5) + c_2(1) = 0\). This simplifies to \(5c_1 + c_2 = 0\).

Step 6: Analyze Solutions for Constants

Now, re-write the simplified equation as \(5c_1 + c_2 = 0\). Consider specific values: set \(c_3=0\), choose \(c_2 = -5\), and \(c_1 = 1\) to satisfy the equation. The existence of such non-trivial solution \((1, -5, 0)\) shows linear dependence.

Key Concepts

Functions

In mathematics, functions describe the relationship between sets of inputs and outputs, typically in terms of mathematical formulas. Each function takes an input value, often represented by a variable like "x," and gives a corresponding output. In the given problem, we have three specific functions:

• \(f_1(x) = 5\): This is a constant function, meaning the output is always 5, regardless of the input value of "x."
• \(f_2(x) = \cos^2 x\): This function involves the cosine of "x," squared. It varies as the angle "x" changes.
• \(f_3(x) = \sin^2 x\): Similarly, this function involves the sine of "x," squared.

To determine if these functions are linearly dependent or independent, we need to find if a non-trivial combination of these gives zero for all "x." This exploration of functions helps us better understand how different mathematical expressions relate to each other.

Trigonometric Identity

Trigonometric identities are useful relationships between trigonometric functions, which make calculations simpler and help prove mathematical statements. One key identity is the Pythagorean trigonometric identity \(\cos^2 x + \sin^2 x = 1\).This identity holds true for any angle "x." It forms the basis of understanding how the sum of squares of sine and cosine equals one, which is essential in various mathematical proofs and problems.In our exercise, this trigonometric identity allows us to deduce that when we add \(\cos^2 x\) and \(\sin^2 x\), we always return the value of 1. This significantly simplifies our problem and helps determine if the given set of functions can be expressed as a non-trivial linear dependence or not.

Linear Combination

A linear combination in mathematics means taking several elements, multiplying them by constants, and then adding the results together. In the context of functions, a linear combination involves different functions, each multiplied by a constant, then summed.In the problem at hand, we form a linear combination as follows:\(c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0\)where \(c_1, c_2,\) and \(c_3\) are constants we are trying to discover.In this problem, substituting the trigonometric identity \(\cos^2 x + \sin^2 x = 1\) into the combination allows us to focus on a simpler expression:\(5c_1 + c_2 = 0\)This approach highlights that it's possible to find constants like \(c_1 = 1\) and \(c_2 = -5\) that satisfy this equation, while \(c_3\) remains zero. The result demonstrates these functions are linearly dependent, as non-zero constants satisfy the linear combination equation.