Problem 11
Question
\((a)\) Derive the result (92) of \(9-6(c)\). (b) Use the Schmidt process to show that a set of three linear combinations of \(1, x, y\) which form an orthogonal set with respect to the weight function \(\sigma=1\) in the domain \(0 \leqq x \leqq 1,0 \leqq y \leqq 1\) is \(v_{1}=1, v_{2}=x-\frac{1}{2}, v_{3}=y-\frac{1}{2}\) (c) Show that the Schmidt orthogonalization process is in general not unique in its determination of the \(N\) orthogonal linear combinations. (d) Prove that a set \(\phi_{1}, \phi_{2}, \ldots, \phi_{N}\) of orthogonal functions is linearly independent.
Step-by-Step Solution
Verified Answer
a) -551, b) It's verified that \(v_{1}=1, v_{2}=x-\frac{1}{2}, v_{3}=y-\frac{1}{2}\) using the Schmidt process. c) The Schmidt process is generally not unique because the order of vectors and their normalization can alter the final result. d) The set \(\phi_{1}, \phi_{2}, \ldots, \phi_{N}\) is linearly independent as no function in the set can be written as a linear combination of the others.
1Step 1: Arithmetic calculation
To find the result of the expression \(9-6(c)\), replace \(c\) with the given number 92 and perform the operation.
2Step 2: Application of Schmidt process
Schmidt orthogonalization process is a method to construct an orthogonal set from an arbitrary set of vectors. The given set {1,x,y} is orthogonalized as follow: \(\) starting with \(v_{1}=1\), the second vector \(v_{2}\) is found by subtracting from \(x\) the projection of \(x\) onto 1, while \(v_{3}\) is found by subtracting from \(y\) the projection of \(y\) onto \(v_{1}\) and \(v_{2}\). The projections are calculated with respect to the weight function \(\sigma=1\).
3Step 3: Non-uniqueness of Schmidt orthogonalization
The Schmidt orthogonalization process is not unique because the orthogonal vectors can be normalized differently, and the order in which the vectors are processed can alter the resulting orthogonal set.
4Step 4: Linear independence of orthogonal functions
A set \(\phi_{1}, \phi_{2}, \ldots, \phi_{N}\) of orthogonal functions is linearly independent because the zero vector cannot be written as a linear combination of these functions, other than the trivial solution, where all coefficients are zero. Hence, no function from this set can be written as a linear combination of the remaining functions.
Key Concepts
Linear AlgebraOrthogonal FunctionsLinear Independence
Linear Algebra
Linear algebra is a fundamental area of mathematics that deals with vectors, matrices, and linear transformations. It provides the tools to solve systems of linear equations and understand vector spaces. In the context of the given exercise, we use linear algebra to express functions like polynomials in terms of basis vectors. These basis vectors allow us to perform operations such as the subtraction of projections to find orthogonal vectors. The Schmidt orthogonalization process is a practical application of linear algebra that constructs orthogonal and orthonormal sets from non-orthogonal vectors. This process is crucial for simplifying vector manipulations and ensuring numerical stability in calculations.
Understanding concepts like inner product (projection) and spanning sets help us to distinguish between orthogonal, orthonormal, and linearly independent sets. Thus, linear algebra not only supports theoretical concepts but also enables the application of these concepts in computational mathematics.
Understanding concepts like inner product (projection) and spanning sets help us to distinguish between orthogonal, orthonormal, and linearly independent sets. Thus, linear algebra not only supports theoretical concepts but also enables the application of these concepts in computational mathematics.
Orthogonal Functions
Orthogonal functions are a set of functions that satisfy the condition of orthogonality under a given inner product. This means that the inner product of any two different functions in the set is zero, indicating that they are independent in some vector space sense. In this exercise, the functions are represented by vectors such as \(1, x, y\), which are made orthogonal using the Schmidt process.
Orthogonal functions are important in many areas, including Fourier series and quantum mechanics, as they simplify the process of decomposing complex functions into simpler, independent components. With respect to the weight function \(\sigma=1\) over the domain \(0 \leqq x \leqq 1, 0 \leqq y \leqq 1\), the orthogonalization yields functions \(v_{1}=1, v_{2}=x-\frac{1}{2}, v_{3}=y-\frac{1}{2}\). This transformation not only simplifies analysis but also aids in numerical computations where orthogonal bases are preferred for stability and simplicity.
In practical terms, orthogonal functions facilitate straightforward calculations since the dot product simplifies to either zero or a constant, reducing computational complexity.
Orthogonal functions are important in many areas, including Fourier series and quantum mechanics, as they simplify the process of decomposing complex functions into simpler, independent components. With respect to the weight function \(\sigma=1\) over the domain \(0 \leqq x \leqq 1, 0 \leqq y \leqq 1\), the orthogonalization yields functions \(v_{1}=1, v_{2}=x-\frac{1}{2}, v_{3}=y-\frac{1}{2}\). This transformation not only simplifies analysis but also aids in numerical computations where orthogonal bases are preferred for stability and simplicity.
In practical terms, orthogonal functions facilitate straightforward calculations since the dot product simplifies to either zero or a constant, reducing computational complexity.
Linear Independence
Linear independence is a vital concept in understanding vector spaces and their bases. A set of vectors (or functions) is linearly independent if no vector in the set can be written as a linear combination of the others, except by using zero coefficients. In step 4 of the solution, we show that a set \(\phi_{1}, \phi_{2}, \ldots, \phi_{N}\) of orthogonal functions is necessarily linearly independent. This is because the zero vector cannot be constructed nontrivially from them, reinforcing the concept that each function carries unique information.
Linear independence is crucial for verifying that a set forms a valid basis for a vector space, allowing for each vector to be uniquely represented. This aspect makes computations efficient, reduces redundancy, and ensures each dimension is accounted for in a representation. For practical applications, understanding linear independence is fundamental in fields like computer graphics, data science, and machine learning, where data integrity and uniqueness play essential roles.
Linear independence is crucial for verifying that a set forms a valid basis for a vector space, allowing for each vector to be uniquely represented. This aspect makes computations efficient, reduces redundancy, and ensures each dimension is accounted for in a representation. For practical applications, understanding linear independence is fundamental in fields like computer graphics, data science, and machine learning, where data integrity and uniqueness play essential roles.
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