Problem 2 Prove that if \(\left\\{\phi_{0}... [FREE SOLUTION]

Chapter 5: Problem 2

Prove that if $\left\\{\phi_{0}, \phi_{1}, \ldots, \phi_{n}\right\\}$ is a set of orthogonal functions, then they must be linearly independent.

Short Answer

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Question: Prove that if a set of functions is orthogonal, it must be linearly independent. Answer: To prove this, we start with a linear combination of orthogonal functions and take the inner product with each function separately. Then, by utilizing the linearity of the inner product and the property of orthogonality, we can show that all the coefficients in the linear combination must be zero, which satisfies the condition for linear independence. Thus, it is proven that a set of orthogonal functions is linearly independent.

Step by step solution

Definition of Orthogonal Functions

Two functions, $\phi_j$ and $\phi_k$, are said to be orthogonal with respect to the given inner product if their inner product is zero, i.e., $\langle\phi_j,\phi_k\rangle = 0.$ In this exercise, a set of $n$ functions $\left\\{\phi_{0}, \phi_{1}, \ldots, \phi_{n}\right\\}$ is said to be orthogonal if any two functions in the set are orthogonal, i.e. $\langle\phi_j,\phi_k\rangle = 0$ for all $j \neq k$.

Definition of Linear Independence

A set of functions $\left\\{\phi_{0}, \phi_{1}, \ldots, \phi_{n}\right\\}$ is said to be linearly independent if no linear combination of them can produce a zero function, except when all the coefficients are zero. In mathematical terms, for constants $c_0, c_1, \ldots, c_n,$ if $$ c_0\phi_0(x) + c_1\phi_1(x) + \cdots + c_n\phi_n(x) = 0 $$ for all $x$, then this implies that $c_0 = c_1 = \cdots = c_n = 0.$

Starting with a Linear Combination

To prove the given set of orthogonal functions is linearly independent, we will start with a linear combination of these functions: $$ c_0\phi_0(x) + c_1\phi_1(x) + \cdots + c_n\phi_n(x) = 0 $$ for all $x$ in the function domain.

Taking the Inner Product with $\phi_i$

Next, we will take the inner product of the given linear combination with each function $\phi_i$ separately, i.e., $0 \leq i \leq n$. For any $i$, we have $$ \langle c_0\phi_0 + c_1\phi_1 + \cdots + c_n\phi_n, \phi_i \rangle = \langle 0, \phi_i \rangle. $$

Using the Linearity of Inner Product

The inner product is linear. Thus, we can rewrite the equation above as $$ c_0\langle\phi_0, \phi_i\rangle + c_1\langle\phi_1,\phi_i\rangle + \cdots + c_n\langle\phi_n, \phi_i\rangle = \langle 0, \phi_i \rangle. $$

Utilizing Orthogonality

We know that the functions are orthogonal, i.e., $\langle\phi_j,\phi_i\rangle = 0$ for all $j \neq i$. Therefore, for any $i$, the inner product of all functions with $\phi_i$, except $\phi_i$ itself, will be zero. This leaves us with the following equation: $$ c_i\langle\phi_i, \phi_i\rangle = \langle 0, \phi_i \rangle. $$

Proving Linear Independence

Since the inner product of a function with itself is non-zero for non-zero functions, i.e., $\langle\phi_i,\phi_i\rangle \neq 0$, this implies that $c_i=0$. We can conclude this for $0 \leq i \leq n$, which means all the coefficients must be zero. Thus, the given set of orthogonal functions is linearly independent.

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Key Concepts

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

Linear Independence

Linear independence is a fundamental concept in mathematics, especially in linear algebra. It refers to a set of functions or vectors where no element in the set can be written as a linear combination of the others. In simpler terms, each function or vector adds something new and unique.

To illustrate, consider a set of functions $\{\phi_0, \phi_1, \dots, \phi_n\}$. These are linearly independent if the equation $c_0\phi_0 + c_1\phi_1 + \cdots + c_n\phi_n = 0$ is only true when all coefficients $c_0, c_1, \ldots, c_n$ are zero. This ensures that no function in the set is redundant, making them essential in vector spaces.

Inner Product

The inner product is a way to multiply two functions or vectors, resulting in a single scalar (a real number). It's a tool used to measure the 'angle' and 'length' between elements in a space. In our exercise above with functions $\phi_j$ and $\phi_k$, the inner product is denoted as $\langle \phi_j, \phi_k \rangle$.

The inner product also helps define orthogonality. If $\langle \phi_j, \phi_k \rangle = 0$, the elements are orthogonal, meaning they are at right angles to each other in the space. This property is crucial for simplifying many problems in linear algebra and functional analysis.

Orthogonality

Orthogonality is a concept that extends the idea of perpendicular vectors to function spaces. When two functions or vectors are orthogonal, their inner product equals zero: $\langle \phi_j, \phi_k \rangle = 0$ for $j eq k$.

Why is this important? Orthogonal functions are inherently linearly independent, simplifying computations in many applications, like Fourier series, quantum mechanics, etc. They allow us to separate complex problems into simpler parts that can be solved independently. This key property comes into play significantly when proving linear independence, as seen in the solution.

Linear Algebra

Linear algebra is the branch of mathematics that studies vectors, vector spaces, and linear transformations. It forms the backbone of numerous areas in math and science, providing tools for solving systems of linear equations, transformations, etc.

In the context of our exercise, linear algebra provides the framework for understanding concepts like linear independence, orthogonality, and inner products. By applying these concepts, linear algebra can help solve equations and analyze function spaces efficiently. It's an essential subject for data science, computer graphics, engineering, and more, as it enables us to model and solve complex, real-world problems.

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Prove that if \(\left\\{\phi_{0}, \phi_{1}, \ldots, \phi_{n}\right\\}\) is a set of orthogonal functions, then they must be linearly independent.

Short Answer

Step by step solution

Definition of Orthogonal Functions

Definition of Linear Independence

Starting with a Linear Combination

Taking the Inner Product with \(\phi_i\)

Using the Linearity of Inner Product

Utilizing Orthogonality

Proving Linear Independence

Key Concepts

Linear Independence

Inner Product

Orthogonality

Linear Algebra

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