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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What is an inner product space?
💡 Hint: Think about how it measures angles and lengths.
Question 2
Easy
Define orthogonality in the context of vector spaces.
💡 Hint: Relate this to geometry, where what does it mean to be at a right angle?
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 is the property of the Cauchy-Schwarz inequality?
- |⟨u,v⟩| ≤ ||u|| + ||v||
- |⟨u,v⟩| ≤ ||u||·||v||
- ⟨u,v⟩ = ||u|| ||v||
💡 Hint: Think about relationships in geometry.
Question 2
True or False: In an inner product space, if |⟨u,v⟩| is greater than ||u||·||v||, then u and v are linearly independent.
- True
- False
💡 Hint: Recall the definition of linear dependence.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Prove that if two vectors are orthogonal in an inner product space, their linear combination with appropriate coefficients will also remain in the span of those vectors.
💡 Hint: Use the properties of linear combinations.
Question 2
Using the Gram–Schmidt process, convert the vectors u=(1,1,1) and v=(1,2,3) into orthonormal basis vectors.
💡 Hint: Apply the iterative projection formulas carefully.
Challenge and get performance evaluation