In this article, we delve into the fascinating world of linear algebra and explore how to determine the linear independence of functions within a vector space. Specifically, we will examine the functions sin x and cos x to determine if they are linearly independent or linearly dependent within the vector space of all functions defined on R. We will follow a rigorous mathematical approach, including a detailed analysis to show that these functions are indeed linearly independent.

Introduction

A key concept in linear algebra is the notion of linear independence. When dealing with functions in a vector space, this means that no function can be expressed as a linear combination of the others. In other words, the functions sin x and cos x are linearly independent if there do not exist non-zero constants c_1 and c_2 such that c_1 sin x c_2 cos x 0 for all x.

Preliminary Definitions

A set of functions is said to be linearly dependent if there exists a non-trivial linear combination of these functions that equals zero. Otherwise, they are said to be linearly independent. In our case, we aim to prove that sin x and cos x are linearly independent.

Analysis and Proof of Linear Independence

To determine whether sin x and cos x are linearly independent, we start by assuming the contrary. Suppose there exist non-zero constants c_1 and c_2 such that the equation c_1 sin x c_2 cos x 0 holds for all x.

Assume without loss of generality that both c_1 and c_2 are non-zero. Without loss of generality, we can assume c_1 > 0 (if not, we can replace c_1 with -c_1 and c_2 with -c_2). Dividing both sides of the equation by c_1, we get:

sin x k cos x 0, where k frac{c_2}{c_1}.

Rearranging, we have:

sin x -k cos x.

Now, we can divide both sides by cos x (assuming cos x eq 0):

tan x -k

This implies:

k -tan x

However, -tan x is not a constant for all x. Since k must be a fixed constant, it cannot satisfy this equation for all x. Therefore, our initial assumption that both c_1 and c_2 are non-zero must be false.

Therefore, either c_1 or c_2 must be zero. Without loss of generality, assume c_1 0. Then we have:

c_2 cos x 0

This equation holds for all x only if c_2 0. Similarly, if c_2 0, then c_1 x 0 must hold for all x, which again implies c_1 0.

Thus, the only solution is c_1 0 and c_2 0, proving that sin x and cos x are linearly independent.

Conclusion

In summary, we have rigorously shown that the functions sin x and cos x are linearly independent within the vector space of all functions defined on R. This conclusion is based on the fact that no non-trivial linear combination of these functions can equal zero for all x.

Understanding the linear independence of functions is crucial in many areas of mathematics and its applications, including differential equations, Fourier analysis, and signal processing. By employing a detailed and logical argument, we have demonstrated that these fundamental trigonometric functions behave as expected, contributing to the richness and depth of mathematical concepts.