Problem 14 - Advanced Engineering Mathematics

Question

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). All functions \(f\) such that \(f(-x)=f(x)\)

Step-by-Step Solution

Answer

The set is a subspace of the vector space \( C(-\infty, \infty) \).

Step 1: Understanding the Problem

We need to determine if the set of functions where every function \( f \) satisfies the property \( f(-x) = f(x) \) forms a subspace of the vector space \( C(-\infty, \infty) \). A subspace must satisfy three conditions: it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication.

Step 2: Check if the Zero Function is Included

The zero function, \( f(x) = 0 \), must be in the set for it to be a subspace. Check if \( f(-x) = f(x) \) holds for \( f(x) = 0 \).Since \( f(x) = 0 \), \( f(-x) = 0 = f(x) \). Thus, the zero function is included in the set.

Step 3: Check for Closure Under Addition

Suppose \( f \) and \( g \) are any two functions in the set where \( f(-x) = f(x) \) and \( g(-x) = g(x) \). We need to show that \( (f + g)(-x) = (f + g)(x) \).Calculate:\[(f + g)(-x) = f(-x) + g(-x) = f(x) + g(x) = (f + g)(x)\]So, the set is closed under addition.

Step 4: Check for Closure Under Scalar Multiplication

For any scalar \( c \) and function \( f \) in the set, verify that \( (cf)(-x) = (cf)(x) \), where \( f(-x) = f(x) \).Calculate:\[(cf)(-x) = c f(-x) = c f(x) = (cf)(x)\]Therefore, the set is closed under scalar multiplication.

Step 5: Conclusion

Since the set contains the zero function, and is closed under both vector addition and scalar multiplication, the set of all functions \( f \) such that \( f(-x) = f(x) \) is a subspace of the vector space \( C(-\infty, \infty) \).

Key Concepts

Subspaces

A subspace is a special kind of subset within a vector space. For a subset to qualify as a subspace, it must meet specific criteria. First, it must include the zero vector. In our exercise, the zero function, given by \( f(x) = 0 \), satisfies \( f(-x) = f(x) \) for all \( x \). Thus, it assures that the zero function is part of the set.

• Zero Vector: Essential for the subset to be a valid subspace. Here, it means the zero function must be present in the set.
• Closure Under Addition: If you take any two functions from the set, their sum must also be in the set. Mathematically, if \( f(-x) = f(x) \) and \( g(-x) = g(x) \), then \((f + g)(-x) = (f + g)(x)\).
• Closure Under Scalar Multiplication: If you take any function from the set and multiply it by a scalar, the resulting function must still belong to the set. This means \((cf)(-x) = (cf)(x)\) for any scalar \( c \).

When these conditions are satisfied, as they are in this exercise, the set is a subspace, making it a crucial concept in studies of linear algebra and vector spaces.

Functional Analysis

Functional Analysis is a branch of mathematics that deals with function spaces and their properties. In this exercise, we are working with a space of continuous functions \( C(-\infty, \infty) \), which is a typical example of a function space you will encounter in functional analysis.Why is this important?

• Function Spaces: These are collections of functions that share a common domain and satisfy particular rules, like continuity or symmetry (e.g., \( f(-x) = f(x) \)).
• Operations and Transformations: Studying such spaces allows for understanding possible transformations and operations that can be performed within these set boundaries.
• Applications: Functional analysis is widely applied in solving differential equations and modeling physical systems.

In summary, functional analysis offers a framework to analyze these function spaces rigorously, helping us solve complex mathematical problems.

Closure Properties

Closure properties are essential in defining the structure of a vector space or a subspace. Essentially, these properties ensure that when operations are performed on members of a set, the results remain within the same set.Closure Under AdditionIf two functions from the set are added, the sum should be such that it satisfies the original condition \( f(-x) = f(x) \). This ensures that when functions are added, they still exhibit the symmetry property.Closure Under Scalar MultiplicationIf a function in the set is multiplied by any scalar, the resulting function must still belong to the set and satisfy \( f(-x) = f(x) \). This ensures that the symmetry property is preserved, even when the function is scaled.These closure properties make it certain that no matter how we combine or modify the functions within a given set using these operations, they will remain part of that set. This is pivotal in proving that a subset is indeed a subspace of a vector space.