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6. Linear Transformations

Definition, Examples, Matrix Representation, Kernel, Image, Properties (Injectivity, Surjectivity)

Linear transformations are fundamental in linear algebra, providing a bridge between vector spaces and matrices. This tutorial covers:


1. Definition of a Linear Transformation

A linear transformation (or linear map) between two vector spaces \( V \) and \( W \) over a field \( \mathbb{F} \) (usually \( \mathbb{R} \) or \( \mathbb{C} \)) is a function \( T: V \to W \) that satisfies:

  1. Additivity:
    [ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \quad \forall \mathbf{u}, \mathbf{v} \in V ]
  2. Homogeneity (Scalar Multiplication):
    [ T(c \mathbf{v}) = c T(\mathbf{v}) \quad \forall c \in \mathbb{F}, \mathbf{v} \in V ]

Key Idea: Linear transformations preserve vector addition and scalar multiplication.


2. Examples of Linear Transformations

(a) Zero Transformation

(b) Identity Transformation

(c) Rotation in \( \mathbb{R}^2 \)

(d) Differentiation (Polynomial Space)

(e) Matrix Multiplication


3. Matrix Representation of Linear Transformations

Every linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^m \) can be represented by an \( m \times n \) matrix \( A \) such that:
[ T(\mathbf{x}) = A\mathbf{x} ]

How to Find the Matrix \( A \)

  1. Apply \( T \) to the standard basis vectors \( \mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n \) of \( \mathbb{R}^n \).
  2. The columns of \( A \) are \( T(\mathbf{e}_1), T(\mathbf{e}_2), \dots, T(\mathbf{e}_n) \).

Example:
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by:
[ T \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 2x + y \ x - 3y \end{pmatrix} ]


4. Kernel (Null Space) and Image (Range)

(a) Kernel (Null Space)

The kernel of \( T \) is the set of all vectors in \( V \) that map to \( \mathbf{0} \):
[ \ker(T) = { \mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0} } ]

Properties:

Example:
For \( T(\mathbf{x}) = A\mathbf{x} \) where \( A = \begin{pmatrix} 1 & 2 \ 3 & 6 \end{pmatrix} \),
[ \ker(T) = \text{Span} \left{ \begin{pmatrix} -2 \ 1 \end{pmatrix} \right} ]

(b) Image (Range)

The image of \( T \) is the set of all outputs in \( W \):
[ \text{Im}(T) = { T(\mathbf{v}) \mid \mathbf{v} \in V } ]

Properties:

Example:
For \( T(\mathbf{x}) = A\mathbf{x} \) where \( A = \begin{pmatrix} 1 & 0 \ 0 & 1 \ 1 & 1 \end{pmatrix} \),
[ \text{Im}(T) = \text{Span} \left{ \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix} \right} ]


5. Properties: Injectivity and Surjectivity

(a) Injectivity (One-to-One)

A linear transformation \( T \) is injective if:
[ T(\mathbf{u}) = T(\mathbf{v}) \implies \mathbf{u} = \mathbf{v} ]
Test:

(b) Surjectivity (Onto)

A linear transformation \( T \) is surjective if:
[ \forall \mathbf{w} \in W, \exists \mathbf{v} \in V \text{ such that } T(\mathbf{v}) = \mathbf{w} ]
Test:

(c) Rank-Nullity Theorem

For \( T: V \to W \),
[ \dim(V) = \dim(\ker(T)) + \dim(\text{Im}(T)) ]

Example:
If \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) has \( \dim(\ker(T)) = 1 \), then \( \dim(\text{Im}(T)) = 2 \).


Summary

| Concept | Definition | Key Property | |———|————|————–| | Linear Transformation | \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \) and \( T(c\mathbf{v}) = cT(\mathbf{v}) \) | Preserves linear structure | | Matrix Representation | \( T(\mathbf{x}) = A\mathbf{x} \) where columns of \( A \) are \( T(\mathbf{e}_i) \) | Encodes transformation | | Kernel | \( \ker(T) = { \mathbf{v} \mid T(\mathbf{v}) = \mathbf{0} } \) | Measures injectivity | | Image | \( \text{Im}(T) = { T(\mathbf{v}) \mid \mathbf{v} \in V } \) | Measures surjectivity | | Injectivity | \( \ker(T) = { \mathbf{0} } \) | One-to-one | | Surjectivity | \( \text{Im}(T) = W \) | Onto |


Practice Problems

  1. Find the matrix representation of \( T(x, y, z) = (2x - y, y + 3z) \).
  2. Determine if \( T(\mathbf{x}) = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} \mathbf{x} \) is injective/surjective.
  3. Compute \( \ker(T) \) and \( \text{Im}(T) \) for \( T(x, y) = (x + y, 2x - y, x) \).

This tutorial covers the core concepts of linear transformations as typically taught in an undergraduate linear algebra course. Let me know if you need further clarification or additional examples!


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