Finding an Orthogonal Basis of Polynomials Using Gram-Schmidt Process

In summary, we use the Gram-Schmidt orthogonalization process to find an orthogonal basis of Span(f1, f2, f3, f4) by defining w1, w2, w3, and w4 as linear combinations of f1, f2, f3, and f4. Starting with g(x) = 1 is a valid approach.
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Homework Statement



Let P2 denote the space of polynomials in k[x] and degree < or = 2. Let f, g exist in P2 such that

f(x) = a2x^2 + a1x + a0
g(x) = b2x^2 + b1x + b0

Define

<f, g> = a0b0 + a1b1 + a2b2

Let f1, f2, f3, f4 be given as below

f1 = x^2 + 3
f2 = 1 - x
f3 = 2x^2 + x + 1
f4 = x + 1

Find an orthogonal basis of Span(f1, f2, f3, f4).

Homework Equations



Gram-Schmidt orthogonalization process.

The Attempt at a Solution



Span(f1, f2, f3, f4) = Span(w1, w2, w3, w4)

Take S = {f, 1}

w1 = f1
w2 = 1 - (<1, f1>/<f1, f1>)*f1
w3 = 1 - (<1, f2>/<f2, f2>)*f2 - (<1, f1>/<f1, f1>)*f1
w4 = 1 - (<1, f3>/<f3, f3>)*f3 - (<1, f2>/<f2, f2>)*f2 - (<1, f1>/<f1, f1>)*f1

Is this the correct procedure? Can I take g = 1 like that?
 
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  • #2
Looks good to me. There's nothing wrong with starting with g(x) = 1.
 
  • #3
Mark44 said:
Looks good to me. There's nothing wrong with starting with g(x) = 1.

Thanks a bunch!
 

1. What is an orthogonal basis?

An orthogonal basis is a set of vectors in a vector space that are all perpendicular (or orthogonal) to each other. This means that the dot product of any two vectors in the basis is equal to 0. Orthogonal bases are commonly used in linear algebra and are important for many mathematical and scientific applications.

2. Why is it important to find an orthogonal basis?

Finding an orthogonal basis allows us to simplify calculations and make them more efficient. It also helps us to understand the structure and relationships between vectors in a vector space. Additionally, orthogonal bases have many useful properties, such as being easier to work with for solving systems of linear equations.

3. How do you find an orthogonal basis?

There are multiple methods for finding an orthogonal basis, depending on the context and specific problem. One method is the Gram-Schmidt process, which involves taking a set of linearly independent vectors and transforming them into an orthogonal basis. Another method is using the QR decomposition, which decomposes a matrix into an orthogonal matrix and an upper triangular matrix.

4. Can an orthogonal basis exist in any vector space?

No, an orthogonal basis can only exist in certain vector spaces. For an orthogonal basis to exist, the vector space must have a defined inner product (also known as a dot product) that satisfies certain properties, such as being symmetric and positive definite.

5. What are some real-world applications of orthogonal bases?

Orthogonal bases have many applications in various fields of science and engineering. They are commonly used in computer graphics and image processing for tasks such as image compression and rotation. They are also used in signal processing, quantum mechanics, and statistics. Additionally, orthogonal bases have applications in areas such as data analysis, optimization, and machine learning.

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