24.10. Rank and Nullity

Try these first

1. 1.

   Define Rank.

   Hint

   Think about linearly independent columns.

2. 2.

   What does Nullity represent?

   Hint

   Relate it to the zeros produced by the matrix.

3. 3.

   What is the rank of a matrix?

   • A) The dimension of the null space
   • B) The dimension of the column space
   • C) The total number of rows

   Hint

   Think about how columns interact in terms of independence.

4. 4.

   True or False: The nullity of a matrix represents the number of linearly independent columns.

   • True
   • False

   Hint

   Focus on the definition of nullity.

5. 5.

   For the following matrix A, determine its rank and nullity. Show your process: A = [[1, 0, 0], [0, 1, 0], [0, 0, 0]].

   Hint

   Identify the leading ones in the rows to determine rank.

6. 6.

   A matrix B has a rank of 3 and 7 columns. Determine its nullity and explain your reasoning.

   Hint

   Use the Rank-Nullity theorem relation.