Chapter 4 Linear Transformations

Outlines Definition and Examples Matrix Representation of linear transformation Similarity

Linear transformations are able to describes.  Translation, rotation & reflection  Solvability of  D x &

Definition: A mapping L from a vector space V into a vector space W is said to be a linear transformation (or a linear operator) if Remark: L is linear

Example 1: Remark: In general, if, the linear transformation can be thought of as a stretching ( ) or shrinking ( ) by a factor of

Example 2:

Example 3:

Example 4:

Example 8:

Example 9 :

Example 10:

Lemma:

Def:

Theorem 4.1.1:

Example 11:

Example 12:

Example 13:

Theorem:

§4.2 Matrix Representations of Linear Transformations Theorem4.2.1:

Proof:

Example: Solution:

Example: Solution:

Figure 4.2.1: (0,1) (1,0) Ax x

Theorem4.2.2:

Example 3: Solution:

Example 4: Solution:

Example 5: Solution:

Theorem 4.2.3

Proof :

Cor : Proof:

Example 6 :

Solution(Method I):

Solution(Method II):

Remark:

Application I : Computer Graphics and Animation Fundamental operators: Dilations and Contractions: Reflection about : e.g., : a reflection about X-axis. : a reflection about Y-axis.

Rotations: Translations: Note: Translation is not linear if Homogeneous Composition of linear mappings is linear!

§4.3 Similarity V W L coordinate mapping (transition matrix)

Question:

Example:

Solution:

Thm 4.3.1

Proof

DEFINITION:

Remark:

Example1:

Solution:

Example2: Solution: