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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
Check if the vectors [1, 2], [2, 4] are linearly independent.
💡 Hint: Look for scalar multiples.
Question 2
Easy
Are the vectors [1, 1] and [2, 2] independent?
💡 Hint: Can you express one in terms of the other?
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 defines linear independence?
- Only trivial solutions exist for linear combinations.
- At least one non-trivial solution exists.
- Vectors are multiples of each other.
💡 Hint: Think about the solution types.
Question 2
True or False: A homogeneous system can have non-trivial solutions.
- True
- False
💡 Hint: Recall the definitions of trivial and non-trivial solutions.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Consider the vectors v_1 = [1, 2, 1], v_2 = [2, 4, 2], and v_3 = [4, 8, 4]. Show that these vectors are linearly dependent.
💡 Hint: Try row-reduction to explore relationships.
Question 2
For the vectors v_1 = [1, 2, 3] and v_2 = [1, a, 3], determine the value(s) of 'a' for which they are independent.
💡 Hint: Set up your system and check the determinant.
Challenge and get performance evaluation