28.4. Kernel and Image of a Linear Transformation

Try these first

1. 1.

   Define the kernel of a linear transformation.

   Hint

   Think about what happens when a transformation sends a vector to zero.

2. 2.

   What is the image of a linear transformation?

   Hint

   Consider all possible outputs of a transformation.

3. 3.

   What does the kernel of a linear transformation represent?

   • Vector space dimension
   • Set of vectors mapped to zero
   • Transformations with non-trivial solutions

   Hint

   Think about the significance of the zero vector.

4. 4.

   True or False: The image of a linear transformation is a subspace of the codomain.

   • True
   • False

   Hint

   Consider how outputs behave under vector space rules.

5. 5.

   Demonstrate the Rank-Nullity Theorem for a transformation T: R^3 -> R^2 defined by T(x, y, z) = (x + y, y + z).

   Hint

   Compute the rank and nullity separately, then add them.

6. 6.

   Given a transformation T: R^3 -> R defined by T(x,y,z) = x + y - z, find the kernel and demonstrate how this relates to the dimension of V.

   Hint

   Set the transformation equation to zero and find the solution set.