23.14. Maximal Linearly Independent Sets

Try these first

1. 1.

   What defines a maximal linearly independent set?

   Hint

   Think about what happens when you add a dependent vector.

2. 2.

   What are pivot columns in a matrix?

   Hint

   Consider where you see the first 1 in each row.

3. 3.

   What is a maximal linearly independent set?

   Hint

   Reflect on the definition of independence.

4. 4.

   True or False: A maximal independent set can include a dependent vector.

   Hint

   Remember the conditions of linear independence.

5. 5.

   Given vectors (1,2,1), (2,4,3), and (3,3,1), find the maximal linearly independent set and explain your method.

   Hint

   Consider how each column contributes to the total span.

6. 6.

   How would you apply the concept of maximal independent sets in a network of sensors? Discuss both potential benefits and pitfalls.

   Hint

   Focus on function versus redundancy.