21.14.3. Kernel and Range

Try these first

1. 1.

   Define the kernel of a linear transformation.

   Hint

   Think about what happens when inputs lead to zero outputs.

2. 2.

   What is the range of a linear transformation?

   Hint

   Consider the images of all input vectors.

3. 3.

   What is the kernel of a linear transformation?

   Hint

   Think about inputs that cause no change.

4. 4.

   The range of T is the set of all vectors mapped to [0] in the codomain. True or False?

   Hint

   Focus on the outputs of the transformation.

5. 5.

   Given a linear transformation T: R^3 → R^3 defined by T(x, y, z) = (x + y - z, 2y, -x + z), find the kernel and range.

   Hint

   Use matrix representation if necessary to analyze.

6. 6.

   How does a change in dimension of the kernel affect the system of equations represented in a transformation?

   Hint

   Relate to the implications of the Rank-Nullity theorem.