26.11. Linear Transformations

Try these first

1. 1.

   Define a linear transformation.

   Hint

   Think about the properties involved in vector addition.

2. 2.

   What is the image of a linear transformation?

   Hint

   What do we call the output set of a function?

3. 3.

   What is a linear transformation?

   • A type of scalar multiplication
   • A function between vector spaces preserving operations
   • Any function mapping numbers

   Hint

   Look back at how it preserves vector operations.

4. 4.

   True or False: The kernel of a linear transformation is always a subspace of the target vector space.

   • True
   • False

   Hint

   Think about where the zero vector maps.

5. 5.

   Given a linear transformation T: R² → R² represented by the matrix A = [[2, 0], [0, 3]], determine the image and kernel of T.

   Hint

   Think about how matrix transformations affect the unit vectors.

6. 6.

   Consider the linear transformation T(x, y) = (x + y, x - y). Is T one-to-one? Justify your answer.

   Hint

   Analyze the equations derived from setting T to zero.