28.6. Composition of Linear Transformations

Try these first

1. 1.

   What is the composition of a linear transformation?

   Hint

   Think about how you would sequence two different actions.

2. 2.

   Write down the formula for composition of T1: U → V and T2: V → W.

   Hint

   What comes first: T1 or T2?

3. 3.

   True or False: The composition of two linear transformations is always a linear transformation.

   • True
   • False

   Hint

   Recall the definition of a linear transformation.

4. 4.

   What is the representation of the composition of T1 and T2 if A is the matrix of T1 and B is the matrix of T2?

   • AB
   • BA
   • A+B

   Hint

   Think about matrix multiplication's order.

5. 5.

   Given two transformations T1(x) = x + 1 and T2(x) = 2x, compute the composition T2 ∘ T1. What does this composition represent mathematically?

   Hint

   Remember to substitute T1 into the expression for T2.

6. 6.

   Prove that if T1 and T2 are linear transformations, then the composition T2 ∘ T1 is also linear by demonstrating it satisfies additivity and homogeneity.

   Hint

   Use the linear properties of T1 and T2 individually to derive the conditions for T2 ∘ T1.