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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Given vectors u = (1, 1) and v = (1, -1), calculate |⟨u,v⟩| and check if it satisfies the Cauchy-Schwarz Inequality.
💡 Hint: Remember to find the norms first.
Question 2
Easy
For u = (2, 2) and v = (1, 3), determine the inner product and norms, checking the Cauchy-Schwarz condition.
💡 Hint: Calculate each component separately.
Practice 4 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
What does the Cauchy-Schwarz Inequality state about two vectors u and v?
- |⟨u,v⟩| ≥ ∥u∥ · ∥v∥
- |⟨u,v⟩| ≤ ∥u∥ · ∥v∥
- |⟨u,v⟩ = ∥u∥ · ∥v∥
💡 Hint: Think about how inner products relate to vector lengths.
Question 2
What condition must hold for equality in the Cauchy-Schwarz Inequality?
- True
- False
💡 Hint: Consider the definition of linear dependence.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Prove that for any vectors u = (u1, u2) and v = (v1, v2) in R², |u1v1 + u2v2| ≤ √(u1² + u2²) * √(v1² + v2²) holds.
💡 Hint: Identify the components systematically for clarity.
Question 2
If a triangle's vertices correspond to vectors A, B, and C, derive the triangle inequality using the Cauchy-Schwarz Inequality.
💡 Hint: Focus on the relationships each side forms with respect to the others.
Challenge and get performance evaluation