21.7. Linear Dependence and Independence

Try these first

1. 1.

   Determine if the following vectors are linearly dependent: (2,4), (1,2), (3,6)

   Hint

   Look for scalar multiples.

2. 2.

   Are the vectors (1, 1) and (1, -1) linearly independent?

   Hint

   Set up the equation and solve.

3. 3.

   What defines linear dependence among vectors?

   • A unique solution exists
   • Multiple solutions exist
   • Only one vector can be expressed as a combination of others

   Hint

   Think in terms of solutions to linear equations.

4. 4.

   True or False: If only linear combinations of zeros result in the zero vector, the vectors are linearly independent.

   • True
   • False

   Hint

   Recall the definitions of dependence versus independence.

5. 5.

   In 3D space, consider vectors A = (1, 0, 0), B = (0, 1, 0), and C = (0, 0, 1). Are these vectors dependent or independent? Justify your answer with a general linear combination.

   Hint

   Use the definition of independence for validation.

6. 6.

   Given the vectors D = (1, 2, 3), E = (2, 4, 6), and F = (1, 1, 1), determine if the set is dependent or independent and explain.

   Hint

   Check for scalar multiples and linear combinations.