25.8. Rank and Nullity Theorem

Try these first

1. 1.

   Define rank in the context of a matrix.

   Hint

   Think about the dimensions of the space these dimensions span.

2. 2.

   What does nullity measure?

   Hint

   Consider what happens when the equation has free variables.

3. 3.

   What does the Rank-Nullity Theorem state?

   • Rank(A) + Nullity(A) = Number of rows
   • Rank(A) + Nullity(A) = Number of columns
   • Rank(A) - Nullity(A) = Number of columns

   Hint

   Think about the dimensions involved in the matrix.

4. 4.

   If a matrix has a rank of 0, what can you say about its nullity?

   • True
   • False

   Hint

   Reflect on how free variables correlate with rank.

5. 5.

   A matrix has 6 columns and a rank of 2. Determine its nullity and discuss what it implies about the solution to the equation Ax=0.

   Hint

   Use the Rank-Nullity theorem to find nullity.

6. 6.

   Consider a system represented by a matrix with rank m. If the system has n variables, derive the implications for the existence of solutions based on the values of m and n.

   Hint

   Reflect on the implications of rank relative to the number of variables.