Practice - Summary Table: Quick Tests for Linear Independence
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Practice Questions
Test your understanding with targeted questions
Determine if the vectors [1, 0] and [0, 1] are linearly independent.
💡 Hint: Check if you can form one vector from the other.
Using the Wronskian, test the independence of functions f(x) = e^x and g(x) = e^(2x).
💡 Hint: Calculate the determinant of the matrix formed by the functions and their derivatives.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the criterion for vectors in Rn to be linearly independent?
💡 Hint: Think about what rank implies for the set of vectors.
Is the Wronskian used for functions or vectors?
💡 Hint: Recall the definition of the Wronskian.
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Challenge Problems
Push your limits with advanced challenges
Given vectors v1 = [1, 2, 3], v2 = [2, 4, 6], and v3 = [1, 0, 1], determine if they form a basis for R3.
💡 Hint: Use row reduction to analyze their relationships.
For the functions f(x) = x and g(x) = x^2, calculate the Wronskian and describe their independence.
💡 Hint: Perform the determinant calculation for completeness.
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