23.3. Definition of Linear Independence

Try these first

1. 1.

   Are the vectors [1, 0] and [0, 1] in R² linearly independent?

   Hint

   Consider their geometric representation.

2. 2.

   Determine if the vectors [1, 1] and [1, 2] are linearly independent.

   Hint

   Look for a solution for coefficients that results in zero.

3. 3.

   What does it mean for a set of vectors to be linearly independent?

   • They span the same space.
   • They can be combined to give zero with non-zero solutions.
   • Their linear combination results in zero only with all coefficients being zero.

   Hint

   Recall the definition given in class.

4. 4.

   Is the set containing the zero vector linearly dependent?

   • True
   • False

   Hint

   Consider the implications of the zero vector.

5. 5.

   Prove that the vectors (1, 1, 1), (2, 2, 2), (3, 3, 3) are linearly dependent by showing how one vector is a linear combination of the others.

   Hint

   Set up the equation and solve using coefficient comparison.

6. 6.

   Given vectors (1, 0) and (a, b) in R², find conditions on a and b for independence.

   Hint

   Use the determinant condition for independence.