By applying the Gram-Schmidt orthogonalization process to the collection {1+t, t, t²}, we obtain the orthonormal collection {u₁, u₂, u₃}, where u₁, u₂, and u₃ are the normalized vectors resulting from the process.
To apply the Gram-Schmidt orthogonalization process and create an orthonormal collection of vectors, let's start with the given collection {1+t, t, t²}.
Step 1: Start with the first vector.
Set v₁ = 1+t.
Step 2: Orthogonalize the second vector.
To orthogonalize the second vector, subtract its projection onto the first vector.
Let's calculate the projection of t onto v₁:
proj(v₁, t) = ⟨t, v₁⟩ / ⟨v₁, v₁⟩ * v₁
= ∫₀₁ t(1+t) dt / ∫₀₁ (1+t)² dt * (1+t)
Simplifying the integral, we have:
proj(v₁, t) = ∫₀₁ (t + t²) dt / ∫₀₁ (1 + 2t + t²) dt * (1+t)
= (1/3 + 1/4) * (1+t)
= (7/12) * (1+t)
Now, we orthogonalize the second vector:
v₂ = t - proj(v₁, t)
= t - (7/12) * (1+t)
Step 3: Orthogonalize the third vector.
To orthogonalize the third vector, subtract its projection onto the first and second vectors.
Let's calculate the projection of t² onto v₁ and v₂:
proj(v₁, t²) = ⟨t², v₁⟩ / ⟨v₁, v₁⟩ * v₁
= ∫₀₁ t²(1+t) dt / ∫₀₁ (1+t)² dt * (1+t)
proj(v₂, t²) = ⟨t², v₂⟩ / ⟨v₂, v₂⟩ * v₂
= ∫₀₁ t² * (t - (7/12) * (1+t)) dt / ∫₀₁ (t - (7/12) * (1+t))² dt * (t - (7/12) * (1+t))
Simplifying the integrals, we have:
proj(v₁, t²) = (1/4 + 1/5) * (1+t)
proj(v₂, t²) = (1/4 - 1/6) * (t - (7/12) * (1+t))
Now, we orthogonalize the third vector:
v₃ = t² - proj(v₁, t²) - proj(v₂, t²)
= t² - (1/4 + 1/5) * (1+t) - (1/4 - 1/6) * (t - (7/12) * (1+t))
Step 4: Normalize the vectors.
To obtain an orthonormal collection, we need to normalize each vector.
Let's normalize v₁, v₂, and v₃:
u₁ = v₁ / ||v₁||
u₂ = v₂ / ||v₂||
u₃ = v₃ / ||v₃||