Introduction to Linear Transformations

Linear transformations are a core concept in linear algebra, providing a way to describe functions that preserve vector addition and scalar multiplication. They are used to model a wide range of systems in mathematics, physics, computer science, and engineering.

Definition

A linear transformation is a function $T$ that maps vectors from one vector space to another, satisfying the following properties:

Additivity:
$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \quad \text{for all vectors } \mathbf{u}, \mathbf{v}.$
Homogeneity:
$T(c\mathbf{u}) = cT(\mathbf{u}) \quad \text{for all vectors } \mathbf{u} \text{ and scalars } c.$

If $T$ satisfies these two properties, it is called a linear transformation.

Matrix Representation

Linear transformations can be represented as matrix-vector multiplication:

$T(\mathbf{x}) = A\mathbf{x},$

where:

$\mathbf{x}$ is the input vector,
$A$ is the transformation matrix,
$T(\mathbf{x})$ is the output vector.

The matrix $A$ captures how the transformation acts on the basis vectors of the input vector space.

Examples

Scaling:
A transformation that scales a vector by a factor $k$ :

$T(\mathbf{x}) = k\mathbf{x},$

is represented by the matrix:

$A = kI,$

where $I$ is the identity matrix.
Rotation:
In $\mathbb{R}^2$ , a rotation by an angle $\theta$ is represented by:

$A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}.$
Reflection:
A reflection over the $x$ -axis is represented by:

$A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$

Properties of Linear Transformations

Key properties of linear transformations include:

Kernel: The set of vectors mapped to the zero vector:
$\text{ker}(T) = \{\mathbf{x} \mid T(\mathbf{x}) = \mathbf{0}\}.$
Image (Range): The set of vectors that can be obtained as outputs of the transformation:
$\text{im}(T) = \{T(\mathbf{x}) \mid \mathbf{x} \text{ in the domain}\}.$
Rank-Nullity Theorem:
$\text{rank}(T) + \text{nullity}(T) = \dim(\text{domain}).$

Geometric Interpretation

Linear transformations can be visualized as operations that transform the entire vector space. Examples include:

Stretching or compressing vectors (scaling).
Rotating vectors in a plane.
Reflecting vectors across a line or plane.

Applications

Linear transformations have numerous applications, including:

Graphics: Transforming images using scaling, rotation, and reflection.
Physics: Modeling rotations and transformations in mechanics and quantum systems.
Data Science: Reducing dimensions using principal component analysis (PCA).
Engineering: Analyzing and designing control systems.

Conclusion

Linear transformations provide a powerful framework for analyzing and manipulating vector spaces. By understanding their properties and matrix representations, we can apply them effectively in diverse fields.

Introduction to Linear Transformations