23.5. Algebraic Criterion for Linear Independence

Try these first

1. 1.

   Check if the vectors [1, 2], [2, 4] are linearly independent.

   Hint

   Look for scalar multiples.

2. 2.

   Are the vectors [1, 1] and [2, 2] independent?

   Hint

   Can you express one in terms of the other?

3. 3.

   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.

4. 4.

   True or False: A homogeneous system can have non-trivial solutions.

   • True
   • False

   Hint

   Recall the definitions of trivial and non-trivial solutions.

5. 5.

   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.

6. 6.

   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.