28.1. Definition of a Linear Transformation

Try these first

1. 1.

   What are the two properties of a linear transformation?

   Hint

   Think of how the transformation interacts with vector addition and scalar multiplication.

2. 2.

   Is the function T(x) = 3x a linear transformation? Why?

   Hint

   Check the properties with addition and scalar multiplication.

3. 3.

   Which of the following properties is NOT satisfied by linear transformations?

   • Additivity
   • Homogeneity
   • Non-linearity

   Hint

   Remember the two fundamental properties!

4. 4.

   True or False: The zero transformation is a type of linear transformation.

   • True
   • False

   Hint

   Does it meet the linear transformation criteria?

5. 5.

   Prove that the transformation T: R^2 → R^2 defined by T(x, y) = (2x, 3y) is linear.

   Hint

   Use arbitrary u = (u1, u2) and v = (v1, v2) to simplify your proof.

6. 6.

   Demonstrate that the transformation T given by T(x) = 5x + 2 is NOT linear.

   Hint

   Examine what happens with the constant addition of 2.