21.14.1. Definition

Try these first

1. 1.

   What is a linear transformation?

   Hint

   Think about how vectors behave under addition.

2. 2.

   Define the kernel of a linear transformation.

   Hint

   Focus on what happens to vectors in the mapping.

3. 3.

   What does a linear transformation preserve?

   • Addition only
   • Scalar multiplication only
   • Both addition and scalar multiplication

   Hint

   Look back at the properties defined.

4. 4.

   True or False: The kernel of a linear transformation is the set of all vectors mapped to zero.

   • True
   • False

   Hint

   Remember the definition of kernel.

5. 5.

   Given the linear transformation T: R^3 → R^3 represented by the matrix [[1, 2, 3], [0, 1, 4], [0, 0, 0]], determine the kernel of this transformation.

   Hint

   Set up the equation involving your transformation matrix and find when it results in the zero vector.

6. 6.

   If a linear transformation has a rank of 2 and operates on a 4-dimensional space, what is the dimension of the kernel?

   Hint

   Remember the rank-nullity theorem connects dimensions; you have the rank and domain.