26.8. Rank and Nullity

Try these first

1. 1.

   Define the term 'rank' in the context of a matrix.

   Hint

   Think about linear independence.

2. 2.

   What does nullity represent?

   Hint

   Consider vectors that solve Ax = 0.

3. 3.

   What is the rank of a square matrix if all of its columns are linearly independent?

   • 0
   • 1
   • n

   Hint

   Consider the maximum independence among columns.

4. 4.

   True or False: Nullity can be negative.

   • True
   • False

   Hint

   Dimensions of spaces are always non-negative.

5. 5.

   Given matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], determine the rank and nullity.

   Hint

   Check the relationship between row operations and linear dependence.

6. 6.

   For a 5x4 matrix with 3 non-zero rows, what can you conclude about its nullity?

   Hint

   Use the Rank-Nullity theorem to find the answer.