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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What does it mean for two subspaces to have a zero intersection?
💡 Hint: Think about the implications of overlap between the two spaces.
Question 2
Easy
Define direct sum in your own words.
💡 Hint: Consider the need for uniqueness in representation.
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
For U and W to form a direct sum, what must their intersection contain?
- All vectors
- Only the zero vector
- Any vectors
💡 Hint: Reflect on the definition of intersection in vector spaces.
Question 2
True or False: Every vector in V can be expressed in more than one way as a sum of vectors from U and W in a direct sum.
- True
- False
💡 Hint: Consider what uniqueness implies.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Show that the following subspaces U = span{(1,0)} and W = span{(0,1)} form a direct sum in ℝ².
💡 Hint: Visualize this in the context of the coordinate plane.
Question 2
Given vector spaces V, U, and W as described in an engineering context, prove that V = U ⊕ W captures all design aspects.
💡 Hint: Think about the contributions each subspace makes to the overall design.
Challenge and get performance evaluation