26.7. Row Space, Column Space, and Null Space

Try these first

1. 1.

   Define row space in your own words.

   Hint

   Think of how you can create new vectors using the rows.

2. 2.

   What is the relationship between rank and column space?

   Hint

   Consider what rank signifies in terms of linear independence.

3. 3.

   What is the row space of a matrix A?

   Hint

   Consider how you can create vectors from the rows.

4. 4.

   True or False: The dimension of the null space increases with the number of linearly independent columns.

   • True
   • False

   Hint

   Think about how rank affects null space.

5. 5.

   Given a matrix A = [[2, 4, -2], [1, 3, -1]], find its row space, column space, and null space.

   Hint

   Start by determining the rank and then use it to find the nullity.

6. 6.

   Prove that adding a scalar multiple of one row to another does not change the row space.

   Hint

   Think about how linear combinations work in a vector space.