24.5. Linear Independence

Try these first

1. 1.

   Identify if the vectors (1, 2) and (3, 6) are independent or dependent.

   Hint

   Look for scalar multiples.

2. 2.

   Are the vectors (2,1) and (1,-2) independent?

   Hint

   Check if you can express one in terms of the other.

3. 3.

   What characterizes a set of linearly independent vectors?

   • They can form the zero vector only with zero coefficients
   • They can be expressed as linear combinations of others
   • They span the entire vector space

   Hint

   Think about what it means for vectors to combine to zero.

4. 4.

   True or False: The vectors (1, 2) and (2, 4) are linearly independent.

   • True
   • False

   Hint

   Check if one vector is a multiplier of the other.

5. 5.

   Prove that the vectors (2, 3) and (2, 6) are dependent by showing a scalar relationship.

   Hint

   Form a linear combination to see if you can make zero.

6. 6.

   Given vectors (3, 1) and (1, 3), find a linear combination that shows they are independent or dependent.

   Hint

   Try setting up the equation with scalars to see if it leads to a solution.