26.5. Linear Independence and Dependence

Try these first

1. 1.

   Determine if the vectors v₁ = (1, 2) and v₂ = (-2, -4) are linearly independent.

   Hint

   Look for a scalar multiple.

2. 2.

   Are the vectors v₁ = (0, 1) and v₂ = (1, 0) independent?

   Hint

   Think about their directions.

3. 3.

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

   • They can be expressed in terms of one another
   • They cannot express the zero vector unless all coefficients are zero
   • They always lie on the same line

   Hint

   Think about how combinations of coefficients work.

4. 4.

   True or False: If one vector in a set can be expressed as a linear combination of the others, the set is linearly independent.

   • True
   • False

   Hint

   Review the definitions of dependence and independence.

5. 5.

   Given the vectors (1,2,3), (2,4,6), and (3,6,9), demonstrate whether they are linearly independent or dependent. Prove your answer mathematically.

   Hint

   Get the system of equations formed by their linear combination equating to zero.

6. 6.

   Take the vectors (1,0,0), (0,1,0), and (0,0,1). Show they span R³ and are independent.

   Hint

   Try forming the identity matrix with these as columns.