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Chapter 3: Problem 3

(a) Show that the vector space \(X\) of all real-valued continuous functions on \([-1,1]\) is the direct sum of the set of all even continuous functions and the set of all odd continuous functions on \([-1,1]\). (b) Give examples of representations of \(\mathbf{R}^{3}\) as a direct sum (i) of a subspace and its orthogonal complement, (ii) of any complementary pair of subspaces.

Short Answer

Expert verified
(a) Continuous functions can be uniquely expressed as sums of even and odd functions. (b) Examples include: (i) \( \mathbb{R}^3 = V \oplus V^\perp \) and (ii) \( \mathbb{R}^3 = U \oplus W \).

Step by step solution

01

Define Vector Space Components

Let vector space \( X \) be the set of all continuous functions defined on the interval \([-1, 1]\). Define: \( E \) as the set of even functions where \( f(-x) = f(x) \) for all \( x \), and \( O \) as the set of odd functions where \( f(-x) = -f(x) \). We aim to show that \( X = E \oplus O \).
02

Show Direct Sum Condition

To show \( X = E \oplus O \), we need to prove each function \( f(x) \in X \) can be uniquely written as a sum of an even and an odd function. Define: \[ f_e(x) = \frac{f(x) + f(-x)}{2} \] and \[ f_o(x) = \frac{f(x) - f(-x)}{2} \]. It can be verified that \( f_e(x) \) is even and \( f_o(x) \) is odd. Also, \( f(x) = f_e(x) + f_o(x) \).
03

Prove Uniqueness

Assume there exists another representation \( f(x) = g(x) + h(x) \) with \( g \in E \) and \( h \in O \). Equating \( f_e + f_o \) to \( g + h \) and analyzing at \( x \) and \( -x \) confirm \( g=f_e \) and \( h=f_o \) affirming the uniqueness of the representation. Thus, \( X = E \oplus O \) is proved.
04

R^3 Example with Orthogonal Complement

For \( \mathbb{R}^3 \), consider subspace \( V = \text{span}\{(1, 0, 0)\} \). Its orthogonal complement, \( V^\perp \), consists of vectors \((0, y, z)\). \( \mathbb{R}^3 \) can thus be expressed as \( V \oplus V^\perp \).
05

R^3 Example with Complementary Subspaces

Another example is using vectors \( (1, 0, 0) \) and the plane \( z = x + y \). Define a subspace \( U = \text{span}\{(1, 0, 0)\} \) and \( W \) as the plane \( z = x + y \). \( \mathbb{R}^3 = U \oplus W \) because every vector can be decomposed uniquely into components from \( U \) and \( W \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even Functions
When we talk about **even functions**, we are looking at functions that have a certain symmetry about the y-axis. For a function \( f(x) \) to be even, it needs to satisfy the condition \( f(-x) = f(x) \) for all real numbers \( x \). This means that when you flip the input value from positive to negative (or vice versa), the output of the function doesn't change.

Examples of even functions include:
  • Quadratic functions like \( f(x) = x^2 \)
  • Cosine function, \( f(x) = \cos(x) \)
  • Constant functions, like \( f(x) = c \) where \( c \) is a constant
Understanding even functions is crucial when working within vector spaces, particularly when determining direct sums.
Odd Functions
**Odd functions** exhibit a different type of symmetry compared to even functions. These functions are symmetrical about the origin. Mathematically, a function \( f(x) \) is odd if it meets the condition \( f(-x) = -f(x) \). So, flipping the input value and the output value has the same effect.

Some common odd functions are:
  • The cubic function \( f(x) = x^3 \)
  • Sine function, \( f(x) = \sin(x) \)
  • The linear function \( f(x) = x \)
In vector spaces, especially within the context of finding orthogonal complements or direct sums, recognizing odd functions helps in organizing components of function sets.
Direct Sum
The concept of a **direct sum** is absolutely indispensable in linear algebra and vector space theory. If we say a vector space \( X \) is a direct sum of two subspaces \( E \) and \( O \), denoted as \( X = E \oplus O \), it means that each element in \( X \) can be expressed uniquely as the sum of an element from \( E \) and an element from \( O \).

To prove a direct sum:
  • Every element \( f(x) \) within the space should be decomposable into \( f_e(x) + f_o(x) \), with \( f_e(x) \) from even space \( E \) and \( f_o(x) \) from odd space \( O \).
  • The decomposition must be unique, meaning there's one and only one way to do it.
In our example, defining \( f_e(x) = \frac{f(x) + f(-x)}{2} \) as even and \( f_o(x) = \frac{f(x) - f(-x)}{2} \) as odd helps in achieving this unique expression for any function in \( X \).
Subspaces
A **subspace** is simply a subset of a vector space that in itself is also a vector space under the same vector addition and scalar multiplication. Key properties to determine subspaces include:
  • Non-empty: It must include the zero vector.
  • Closed under addition: The sum of any two elements in the subspace must be within the subspace.
  • Closed under scalar multiplication: Multiplying any element by a scalar must also yield an element in the subspace.
In the case of \( \mathbb{R}^3 \), subspaces could be defined using lines or planes passing through the origin, such as \( \text{span}\{(1, 0, 0)\} \). Note how these are crucial for understanding complementary subspaces and orthogonal complements.
Orthogonal Complements
The **orthogonal complement** of a subspace \( V \) within a vector space \( X \) can be thought of as all the vectors in \( X \) that are perpendicular to every vector in \( V \). If \( X = \mathbb{R}^3 \) and \( V = \text{span}\{(1, 0, 0)\} \), the orthogonal complement \( V^\perp \) consists of vectors of the form \((0, y, z)\).

Properties of orthogonal complements:
  • They complete the vector space, allowing \( X \) to be expressed as \( V \oplus V^\perp \).
  • Orthogonal complements are vital for breaking down multi-dimensional spaces into simpler, understandable components.
  • They provide a standard method for direct sum decompositions, particularly useful in function space problems and matrix theory.
Understanding orthogonal complements enriches problem-solving strategies in vector spaces, offering a powerful tool for decomposing vectors.

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