23.10. Exercises

Try these first

1. 1.

   Determine if the set of vectors { (1,0), (0,1) } is linearly independent.

   Hint

   Consider their geometric representation.

2. 2.

   Is the vector { (1,1), (1,1) } linearly independent?

   Hint

   Look for redundancy in vectors.

3. 3.

   True or False: A set of vectors is linearly independent if at least one vector can be expressed as a combination of others.

   • True
   • False

   Hint

   Recall the definitions of independence and dependence.

4. 4.

   Which of the following sets of vectors are linearly dependent? {(1,0),(1,0)}, {(1,2),(2,4)}, or {(1,1),(1,-1)}?

   • Set A
   • Set B
   • Set C

   Hint

   Look for vectors that can be expressed in terms of each other.

5. 5.

   Given the vectors {(1,2,3), (2,4,6), (3,6,9)}, demonstrate through matrix methods that they are dependent.

   Hint

   Look at the ratios between the rows.

6. 6.

   In R^3, provide a set of four vectors and argue whether they can be independent or dependent.

   Hint

   Recall dimensional limits in linear algebra.