Lecture on Linear Transformations

Introduction to Linear Transformations

• Definition: A special transformation in linear algebra, an extension of the concept of a transformation.
• A transformation is essentially a function, denoted as T: ( \mathbb{R}^n \to \mathbb{R}^m ).

Conditions for a Linear Transformation

1. Additivity:
   • For vectors ( , \textbf{a}, \textbf{b} \in \mathbb{R}^n , ), a transformation T is linear if: [ T(\textbf{a} + \textbf{b}) = T(\textbf{a}) + T(\textbf{b}) ]
2. Scalar Multiplication:
   • For any scalar ( , c , ) and vector ( , \textbf{a} ), T is linear if: [ T(c \cdot \textbf{a}) = c \cdot T(\textbf{a}) ]

Example of Linear Transformation

• Given transformation T:
  • T: ( \mathbb{R}^2 \to \mathbb{R}^2 )
  • ( T(x_1, x_2) = (x_1 + x_2, 3x_1) )

Verifying Linear Transformation

• Additivity Check:

  • For vectors ( \textbf{a} = (a_1, a_2) ) and ( \textbf{b} = (b_1, b_2) ):
    • ( T(\textbf{a} + \textbf{b}) = T((a_1 + b_1, a_2 + b_2)) = (a_1 + a_2 + b_1 + b_2, 3a_1 + 3b_1) )
    • ( T(\textbf{a}) + T(\textbf{b}) = (a_1 + a_2, 3a_1) + (b_1 + b_2, 3b_1) = (a_1 + a_2 + b_1 + b_2, 3a_1 + 3b_1) )
    • Shows ( T(\textbf{a} + \textbf{b}) = T(\textbf{a}) + T(\textbf{b}) )

• Scalar Multiplication Check:

  • For a scalar ( , c , ):
    • ( T(c \cdot \textbf{a}) = T((ca_1, ca_2)) = (ca_1 + ca_2, 3ca_1) )
    • ( c \cdot T(\textbf{a}) = c \cdot (a_1 + a_2, 3a_1) = (ca_1 + ca_2, 3ca_1) )
    • Shows ( T(c \cdot \textbf{a}) = c \cdot T(\textbf{a}) )

Example of Non-Linear Transformation

• Transformation T: ( T(x_1, x_2) = (x_1^2, 0) )

Verifying Non-Linear Transformation

• Scalar Multiplication Check:
  • ( T(c \cdot \textbf{a}) = T((ca_1, ca_2)) = (ca_1)^2 = c^2a_1^2 )
  • ( c \cdot T(\textbf{a}) = c \cdot (a_1^2, 0) = (ca_1^2, 0) )
  • Shows ( T(c \cdot \textbf{a}) \neq c \cdot T(\textbf{a}) ), thus not linear

General Observations

• Linear transformations involve only linear combinations of input components.
• Non-linear transformations involve exponents or products of components.

Conclusion

• Linear transformations are foundational in linear algebra, leading to further concepts and interesting results in subsequent discussions.