28.5. Rank and Nullity

Try these first

1. 1.

   Define 'rank' in the context of linear transformations.

   Hint

   Think about the number of output dimensions.

2. 2.

   What does 'nullity' represent?

   Hint

   Consider how many inputs can be mapped to zero.

3. 3.

   What is the rank of a linear transformation?

   • Dimension of kernel
   • Dimension of image
   • Dimension of domain

   Hint

   It relates directly to the output of the transformation.

4. 4.

   True or False: Nullity refers to the dimension of the image.

   • True
   • False

   Hint

   Remember the definition of each term.

5. 5.

   Consider a linear transformation T: R^5 → R^3 where the nullity is 4. What is the rank?

   Hint

   Use the dimension relationship between the spaces.

6. 6.

   For a linear transformation represented by a matrix B with 6 columns and an image dimension of 3, what is the kernel dimension?

   Hint

   Recall the Rank-Nullity theorem's formula.