Linear Transformations | Generated by AI
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:
- Definition of linear transformations
- Examples of common linear transformations
- Matrix representation of linear transformations
- Kernel (null space) and Image (range)
- Properties: Injectivity (one-to-one) and Surjectivity (onto)
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:
- Additivity:
\[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \quad \forall \mathbf{u}, \mathbf{v} \in V \] - 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
- \( T(\mathbf{v}) = \mathbf{0} \) for all \( \mathbf{v} \in V \).
(b) Identity Transformation
- \( T(\mathbf{v}) = \mathbf{v} \) for all \( \mathbf{v} \in V \).
(c) Rotation in \( \mathbb{R}^2 \)
- Rotating a vector by angle \( \theta \):
\[ T \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
(d) Differentiation (Polynomial Space)
- \( T: P_n \to P_{n-1} \) where \( T(p(x)) = p’(x) \).
(e) Matrix Multiplication
- For a fixed \( m \times n \) matrix \( A \), \( T: \mathbb{R}^n \to \mathbb{R}^m \) is defined by \( T(\mathbf{x}) = A\mathbf{x} \).
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 \( A \)
- Apply \( T \) to the standard basis vectors \( \mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n \) of \( \mathbb{R}^n \).
- 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:
\[
T \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 2x + y \ x - 3y \end{pmatrix}
\]
- Compute \( T(\mathbf{e}_1) = T(1, 0) = (2, 1) \)
- Compute \( T(\mathbf{e}_2) = T(0, 1) = (1, -3) \)
- Thus, the matrix \( A \) is:
\[ A = \begin{pmatrix} 2 & 1 \ 1 & -3 \end{pmatrix} \]
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:
- \( \ker(T) \) is a subspace of \( V \).
- \( T \) is injective (one-to-one) if and only if \( \ker(T) = { \mathbf{0} } \).
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:
- \( \text{Im}(T) \) is a subspace of \( W \).
- \( T \) is surjective (onto) if and only if \( \text{Im}(T) = W \).
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:
- \( T \) is injective \( \iff \ker(T) = { \mathbf{0} } \).
- If \( \dim(V) < \dim(W) \), \( T \) may not be injective.
(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:
- \( T \) is surjective \( \iff \text{Im}(T) = W \).
- If \( \dim(V) > \dim(W) \), \( T \) may not be surjective.
(c) Rank-Nullity Theorem
For \( T: V \to W \),
\[
\dim(V) = \dim(\ker(T)) + \dim(\text{Im}(T))
\]
- Rank \( = \dim(\text{Im}(T)) \)
- Nullity \( = \dim(\ker(T)) \)
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
- Find the matrix representation of \( T(x, y, z) = (2x - y, y + 3z) \).
- Determine if \( T(\mathbf{x}) = \begin{pmatrix} 1 & 2 \ 0 & 1 \end{pmatrix} \mathbf{x} \) is injective/surjective.
- 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!