In this section, we will explore the concept of linear transformations, their definitions, and their fundamental properties. Linear transformations are crucial in various fields, including computer graphics, physics, and engineering, as they provide a mathematical framework for manipulating vectors and points in space.

A linear transformation is a function \( T: V \rightarrow W \) between two vector spaces \( V \) and \( W \) that preserves the operations of vector addition and scalar multiplication. Formally, a function \( T \) is a linear transformation if for all vectors \( \mathbf{u}, \mathbf{v} \in V \) and all scalars \( c \in \mathbb{R} \):

1. Additivity (or Superposition): \[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \]

2. Homogeneity (or Scalar Multiplication): \[ T(c\mathbf{u}) = cT(\mathbf{u}) \]

Scaling is a linear transformation that enlarges or shrinks vectors by a scalar factor. For a vector \( \mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix} \) and a scalar \( k \), the scaling transformation \( T \) is defined as:

\[ T(\mathbf{v}) = k \mathbf{v} = \begin{bmatrix} kx \ ky \end{bmatrix} \]

Rotation is a linear transformation that rotates vectors around the origin by a certain angle \( \theta \). For a vector \( \mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix} \), the rotation transformation \( T \) is defined as:

\[ T(\mathbf{v}) = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \]

Reflection is a linear transformation that flips vectors over a specified axis. For instance, reflecting over the x-axis for a vector \( \mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix} \) is defined as:

\[ T(\mathbf{v}) = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} x \ -y \end{bmatrix} \]

As defined, a linear transformation must satisfy both additivity and homogeneity. This ensures that the transformation respects the structure of the vector space.

Every linear transformation can be represented by a matrix. If \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \) is a linear transformation, there exists an \( m \times n \) matrix \( A \) such that for any vector \( \mathbf{x} \in \mathbb{R}^n \):

\[ T(\mathbf{x}) = A\mathbf{x} \]

The composition of two linear transformations is also a linear transformation. If \( T_1: V \rightarrow W \) and \( T_2: W \rightarrow U \) are linear transformations, then the composition \( T_2 \circ T_1: V \rightarrow U \) is a linear transformation.

A linear transformation \( T: V \rightarrow W \) is invertible if there exists a linear transformation \( T^{-1}: W \rightarrow V \) such that:

\[ T^{-1}(T(\mathbf{v})) = \mathbf{v} \quad \text{for all} \quad \mathbf{v} \in V \]

and

\[ T(T^{-1}(\mathbf{w})) = \mathbf{w} \quad \text{for all} \quad \mathbf{w} \in W \]

Let's consider a practical example in Python to illustrate a linear transformation using matrix representation.

1. We define a vector vector as \([2, 3]\).
2. We define a transformation matrix transformation_matrix as a scaling matrix that scales by a factor of 2.
3. We apply the transformation using the dot product (np.dot) to get the transformed vector.

Given the vector \( \mathbf{v} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \) and the transformation matrix \( A = \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix} \), apply the linear transformation and find the transformed vector.

Prove that the rotation matrix \( R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \) is a linear transformation.

To prove that \( R(\theta) \) is a linear transformation, we need to show that it satisfies additivity and homogeneity.

1. Additivity:

   Let \( \mathbf{u} = \begin{bmatrix} u_1 \ u_2 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix} \).

   \[ R(\theta)(\mathbf{u} + \mathbf{v}) = R(\theta) \begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \end{bmatrix} \]

   \[ = \begin{bmatrix} \cos \theta (u_1 + v_1) - \sin \theta (u_2 + v_2) \ \sin \theta (u_1 + v_1) + \cos \theta (u_2 + v_2) \end{bmatrix} \]

   \[ = \begin{bmatrix} \cos \theta u_1 - \sin \theta u_2 \ \sin \theta u_1 + \cos \theta u_2 \end{bmatrix} + \begin{bmatrix} \cos \theta v_1 - \sin \theta v_2 \ \sin \theta v_1 + \cos \theta v_2 \end{bmatrix} \]

   \[ = R(\theta)(\mathbf{u}) + R(\theta)(\mathbf{v}) \]

2. Homogeneity:

   Let \( c \) be a scalar and \( \mathbf{u} = \begin{bmatrix} u_1 \ u_2 \end{bmatrix} \).

   \[ R(\theta)(c\mathbf{u}) = R(\theta) \begin{bmatrix} cu_1 \ cu_2 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} cu_1 \ cu_2 \end{bmatrix} \]

   \[ = \begin{bmatrix} c(\cos \theta u_1 - \sin \theta u_2) \ c(\sin \theta u_1 + \cos \theta u_2) \end{bmatrix} \]

   \[ = c \begin{bmatrix} \cos \theta u_1 - \sin \theta u_2 \ \sin \theta u_1 + \cos \theta u_2 \end{bmatrix} \]

   \[ = c R(\theta)(\mathbf{u}) \]

Thus, \( R(\theta) \) satisfies both additivity and homogeneity, proving it is a linear transformation.

In this section, we have defined linear transformations and explored their fundamental properties. We have seen how linear transformations can be represented using matrices and how they preserve vector space operations. Understanding these concepts is crucial for manipulating vectors and points in 3D space, which we will delve into further in the upcoming sections.