Question

Show that \(\left\\{y_{1}, y_{2}, \ldots, y_{n}\right\\}\) is linearly dependent on \((a, b)\) if and only if at least one of the functions \(y_{1}, y_{2}, \ldots, y_{n}\) can be written as a linear combination of the others on \((a, b)\).

Short answer

Based on the solution provided, answer the following question: Explain the relationship between linear dependence and the ability to express one function as a linear combination of the others in a set of functions. Answer: The relationship between linear dependence and the ability to express one function as a linear combination of the others is that they are equivalent. If at least one function in a set can be written as a linear combination of the others, it implies that the set of functions is linearly dependent. Conversely, if a set of functions is linearly dependent, it implies that there exists at least one function that can be written as a linear combination of the other functions in the set.

Step-by-step solution

Definition of linear dependence

Linear dependence is a property of a set of vectors where some of the vectors can be written as a linear combination of the others. For a set of functions \(\left\\{y_{1}, y_{2}, \ldots, y_{n}\right\\}\), this means there exist constants \(c_1, c_2, \ldots, c_n\), not all of which are zero, such that $$c_1y_{1}(x) + c_2y_{2}(x) + \ldots + c_ny_{n}(x) = 0$$ for all \(x\) in \((a, b)\).

Prove that if at least one function can be written as a linear combination, the set is linearly dependent.

Assume that at least one of the functions, without loss of generality let it be \(y_{1}\), can be written as a linear combination of the others: $$y_{1}(x) = c_2y_{2}(x) + \ldots + c_ny_{n}(x)$$ for all \(x\) in \((a, b)\), where at least one of the coefficients \(c_2, c_3, \ldots, c_n\) is nonzero. Then, we can rewrite it as a linear dependence relation: $$(-1)y_{1}(x) + c_2y_{2}(x) + \ldots + c_ny_{n}(x) = 0$$ for all \(x\) in \((a, b)\). Since at least one of the coefficients is nonzero, the given set of functions is linearly dependent.

Prove that if the set is linearly dependent, at least one function can be written as a linear combination of the others.

If the set \(\left\\{y_{1}, y_{2}, \ldots, y_{n}\right\\}\) is linearly dependent, there exist constants \(c_1, c_2, \ldots, c_n\), not all of which are zero, such that $$c_1y_{1}(x) + c_2y_{2}(x) + \ldots + c_ny_{n}(x) = 0$$ for all \(x\) in \((a, b)\). Let \(c_j\) be nonzero for some \(j\) in the range \(1 \leq j \leq n\). Then, we can isolate the function \(y_{j}(x)\) by writing: $$y_{j}(x) = -\frac{c_1}{c_j}y_{1}(x) - \frac{c_2}{c_j}y_{2}(x) - \ldots - \frac{c_{j-1}}{c_j}y_{j-1}(x) - \frac{c_{j+1}}{c_j}y_{j+1}(x) - \ldots - \frac{c_n}{c_j}y_{n}(x)$$ for all \(x\) in \((a, b)\). Since \(y_{j}(x)\) can be written as a linear combination of the other functions, the second statement of our claim holds.

Conclusion

We showed that if at least one function can be written as a linear combination of the others, then the set is linearly dependent. We also proved that if the set is linearly dependent, at least one of the functions can be written as a linear combination of the others. Thus, we have demonstrated that the given statement is true.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They appear frequently in various scientific disciplines like physics, engineering, and biology to describe processes that change over time. In essence, they allow us to analyze rates of change and behaviors of dynamic systems. There are several types of differential equations, but the most commonly addressed in beginner courses include:

• Ordinary Differential Equations (ODEs): These involve functions of a single variable and their derivatives.
• Partial Differential Equations (PDEs): These involve functions of multiple variables and their partial derivatives.

Understanding how solutions to differential equations are formed, particularly through linear dependence and independence, is crucial. These solutions often appear as linear combinations of fundamental solutions, making the study of such combinations key in determining the behavior of the system over a given interval.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vector spaces and the linear mappings between them. It's fundamental in various areas of science and engineering. In the context of the given exercise, we are primarily concerned with vectors in the form of functions on an interval \((a, b)\).
Key concepts include:

• Vectors: Objects that can be added together and multiplied (scaled) by numbers, referred to as scalars.
• Vector Spaces: Collections of vectors where linear combinations are defined.
• Linear Combinations: A sum of scalar multiples of vectors.

Understanding the linear dependence of these vector functions is essential. It helps in analyzing whether a set of functions can be expressed in terms of others, which has profound implications in both theory and application.

Linear Combinations

A linear combination involves multiplying vectors by scalars and summing the results. This concept is at the heart of linear dependence and independence. In linear algebra, a set of vectors (or functions, in the context of differential equations) is considered dependent if at least one of the vectors can be expressed as a linear combination of the others.
For example, given a set of functions \(\{y_1, y_2, \ldots, y_n\}\), determining whether a function can be written as:\[y_1(x) = c_2y_2(x) + c_3y_3(x) + \ldots + c_ny_n(x)\]with some non-zero coefficients \(c_2, c_3, \ldots, c_n\) is essential. If such a relationship holds, it signifies linear dependence, indicating redundancy in the set of functions. Linear combinations simplify complex mathematical models by reducing unnecessary components and focusing on the most significant parts.