24.9. Row Space, Column Space, and Null Space

Try these first

1. 1.

   What is the row space of a 2x3 matrix?

   Hint

   Think about what span means.

2. 2.

   How do you determine if a vector is in the column space of a matrix?

   Hint

   Consider using row reduction.

3. 3.

   What does the row space of a matrix represent?

   • All row vectors
   • All linear combinations of row vectors
   • Rank of the matrix

   Hint

   Think about the definition of span.

4. 4.

   True or False: The null space contains vectors that can be expressed as a linear combination of the row space.

   • True
   • False

   Hint

   Consider the definition of null space.

5. 5.

   Given a matrix A with explicit entries, calculate the row space, column space, and null space. Provide a geometric interpretation.

   Hint

   Use row reduction and linear combination analyses.

6. 6.

   Explain how changing one of the rows in matrix A affects the rank and nullity. Include an example with illustrations.

   Hint

   Visualize with a change in basis.