28.3. The Matrix of a Linear Transformation

Try these first

1. 1.

   Define a linear transformation.

   Hint

   Think about operations that must be preserved.

2. 2.

   What is a standard matrix?

   Hint

   How does it relate to the transformation of basis vectors?

3. 3.

   What does the standard matrix of a linear transformation represent?

   • It represents the input vector
   • It provides a unique representation for the transformation
   • It is irrelevant for linear algebra.

   Hint

   Consider how matrices simplify transformations.

4. 4.

   True or False: Every linear transformation can be represented by a matrix.

   • True
   • False

   Hint

   Reflect on the definition of linear transformations.

5. 5.

   Given a linear transformation T: R² → R² that reflects points across the x-axis. Derive the standard matrix for this transformation.

   Hint

   Consider how the reflection across the x-axis affects points.

6. 6.

   Suppose T is a rotation transformation through an angle α. Create a general expression for its standard matrix.

   Hint

   Think about how angles are represented in rotation.