28. Linear Transformations

Try these first

1. 1.

   Define a linear transformation.

   Hint

   Think about what properties must hold for the transformation.

2. 2.

   What is the kernel of a linear transformation?

   Hint

   Consider where the transformation takes vectors.

3. 3.

   What does a linear transformation preserve?

   • Addition and scalar multiplication
   • Only addition
   • Only scalar multiplication

   Hint

   Think about the properties of operations in vector spaces.

4. 4.

   True or False: The kernel of a transformation includes all vectors mapped to zero.

   • True
   • False

   Hint

   Consider what it means to be mapped to zero.

5. 5.

   Given the linear transformation T(x, y) = (2x, 3y), determine the kernel and image.

   Hint

   Evaluate what inputs lead to the zero output to find the kernel.

6. 6.

   If A is a 3x3 matrix with rank 2, what can be said about its nullity?

   Hint

   Use the Rank-Nullity theorem as a guide for this question.