Question 1168385: A)Consider the vector space P2. Define the inner product,
⟨p, q⟩ = ∫(from0 to1) p(x)q(x) dx
Use the Gram-Schmidt process to transform the standard basis S = {1, x, x^2} into an orthonormal basis.
A.1)Express r(x) = 1 + x + 4x^2 as a linear combination of the vectors in the orthonormal basis for P2 found in the previous exercise.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step by step.
**A) Gram-Schmidt Process**
We are given the standard basis S = {1, x, x²} and the inner product ⟨p, q⟩ = ∫(from 0 to 1) p(x)q(x) dx.
1. **v₁ = u₁ = 1**
2. **v₂ = u₂ - proj_v₁(u₂) = x - ⟨x, v₁⟩ / ⟨v₁, v₁⟩ * v₁**
* ⟨x, v₁⟩ = ∫(from 0 to 1) x * 1 dx = [x²/2] from 0 to 1 = 1/2
* ⟨v₁, v₁⟩ = ∫(from 0 to 1) 1 * 1 dx = [x] from 0 to 1 = 1
* v₂ = x - (1/2) / 1 * 1 = x - 1/2
3. **v₃ = u₃ - proj_v₁(u₃) - proj_v₂(u₃) = x² - ⟨x², v₁⟩ / ⟨v₁, v₁⟩ * v₁ - ⟨x², v₂⟩ / ⟨v₂, v₂⟩ * v₂**
* ⟨x², v₁⟩ = ∫(from 0 to 1) x² * 1 dx = [x³/3] from 0 to 1 = 1/3
* ⟨x², v₂⟩ = ∫(from 0 to 1) x² * (x - 1/2) dx = ∫(from 0 to 1) (x³ - x²/2) dx = [x⁴/4 - x³/6] from 0 to 1 = 1/4 - 1/6 = 1/12
* ⟨v₂, v₂⟩ = ∫(from 0 to 1) (x - 1/2)² dx = ∫(from 0 to 1) (x² - x + 1/4) dx = [x³/3 - x²/2 + x/4] from 0 to 1 = 1/3 - 1/2 + 1/4 = 1/12
* v₃ = x² - (1/3) / 1 * 1 - (1/12) / (1/12) * (x - 1/2) = x² - 1/3 - (x - 1/2) = x² - x + 1/6
Now, normalize the vectors:
1. **e₁ = v₁ / ||v₁||**
* ||v₁|| = √⟨v₁, v₁⟩ = √1 = 1
* e₁ = 1 / 1 = 1
2. **e₂ = v₂ / ||v₂||**
* ||v₂|| = √⟨v₂, v₂⟩ = √(1/12) = 1 / √12 = 1 / (2√3)
* e₂ = (x - 1/2) / (1 / (2√3)) = 2√3(x - 1/2) = 2√3x - √3
3. **e₃ = v₃ / ||v₃||**
* ||v₃|| = √⟨v₃, v₃⟩ = √∫(from 0 to 1) (x² - x + 1/6)² dx = √(1/180) = 1 / √(180) = 1 / (6√5)
* e₃ = (x² - x + 1/6) / (1 / (6√5)) = 6√5(x² - x + 1/6) = 6√5x² - 6√5x + √5
Therefore, the orthonormal basis is {1, 2√3x - √3, 6√5x² - 6√5x + √5}.
**A.1) Express r(x) = 1 + x + 4x² as a linear combination of the vectors in the orthonormal basis.**
Let r(x) = c₁e₁ + c₂e₂ + c₃e₃.
* c₁ = ⟨r(x), e₁⟩ = ∫(from 0 to 1) (1 + x + 4x²) * 1 dx = [x + x²/2 + 4x³/3] from 0 to 1 = 1 + 1/2 + 4/3 = 17/6
* c₂ = ⟨r(x), e₂⟩ = ∫(from 0 to 1) (1 + x + 4x²) * (2√3x - √3) dx = ∫(from 0 to 1) (2√3x - √3 + 2√3x² - √3x + 8√3x³ - 4√3x²) dx = ∫(from 0 to 1) (8√3x³ - 2√3x² + √3x - √3) dx = [2√3x⁴ - 2√3x³/3 + √3x²/2 - √3x] from 0 to 1 = 2√3 - 2√3/3 + √3/2 - √3 = (12√3 - 4√3 + 3√3 - 6√3) / 6 = 5√3 / 6
* c₃ = ⟨r(x), e₃⟩ = ∫(from 0 to 1) (1 + x + 4x²) * (6√5x² - 6√5x + √5) dx = ∫(from 0 to 1) (6√5x² - 6√5x + √5 + 6√5x³ - 6√5x² + √5x + 24√5x⁴ - 24√5x³ + 4√5x²) dx = ∫(from 0 to 1) (24√5x⁴ - 18√5x³ + 4√5x² - 5√5x + √5) dx = [24√5x⁵/5 - 18√5x⁴/4 + 4√5x³/3 - 5√5x²/2 + √5x] from 0 to 1 = 24√5/5 - 9√5/2 + 4√5/3 - 5√5/2 + √5 = (144√5 - 135√5 + 40√5 - 75√5 + 30√5) / 30 = 4√5 / 30 = 2√5 / 15
Therefore, r(x) = (17/6) * 1 + (5√3 / 6) * (2√3x - √3) + (2√5 / 15) * (6√5x² - 6√5x + √5).
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