23.17. Summary Table: Quick Tests for Linear Independence

Try these first

1. 1.

   Determine if the vectors [1, 0] and [0, 1] are linearly independent.

   Hint

   Check if you can form one vector from the other.

2. 2.

   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.

3. 3.

   What is the criterion for vectors in Rn to be linearly independent?

   • Row reduced matrix must have full rank
   • Vectors must be orthogonal
   • Vectors cannot be identical

   Hint

   Think about what rank implies for the set of vectors.

4. 4.

   Is the Wronskian used for functions or vectors?

   • True
   • False

   Hint

   Recall the definition of the Wronskian.

5. 5.

   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.

6. 6.

   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.