21.7.1. Definition

Try these first

1. 1.

   Determine whether the vectors (1, 1) and (2, 2) are linearly dependent.

   Hint

   Look for scalar multiples.

2. 2.

   Are the vectors (1, 0) and (0, 1) linearly independent?

   Hint

   Check if one can be created through the other.

3. 3.

   What does it mean for vectors to be linearly dependent?

   • They point in the same direction
   • At least one can be expressed as a combination of others
   • They are orthogonal

   Hint

   Think of how many directions they actually span.

4. 4.

   True or False: A set of vectors can be both linearly dependent and independent at the same time.

   • True
   • False

   Hint

   Consider the definitions.

5. 5.

   Given vectors v₁ = (1, 1, 1), v₂ = (2, 2, 2), and v₃ = (3, 3, 3), prove whether they are dependent or independent.

   Hint

   Look for scalar relationships.

6. 6.

   Using the vectors u = (1, 2, 3), v = (4, 5, 6), and w = (7, 8, 9), find the conditions under which these can be independent.

   Hint

   Matrix properties indicate independence.