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Definition and Properties

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Introduction

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.

What is a Linear Transformation?

A linear transformation is a function between two vector spaces and that preserves the operations of vector addition and scalar multiplication. Formally, a function is a linear transformation if for all vectors and all scalars :

  1. Additivity (or Superposition):

  2. Homogeneity (or Scalar Multiplication):

Examples of Linear Transformations

Example 1: Scaling

Scaling is a linear transformation that enlarges or shrinks vectors by a scalar factor. For a vector and a scalar , the scaling transformation is defined as:

Example 2: Rotation

Rotation is a linear transformation that rotates vectors around the origin by a certain angle . For a vector , the rotation transformation is defined as:

Example 3: Reflection

Reflection is a linear transformation that flips vectors over a specified axis. For instance, reflecting over the x-axis for a vector is defined as:

Properties of Linear Transformations

Property 1: Linearity

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

Property 2: Matrix Representation

Every linear transformation can be represented by a matrix. If is a linear transformation, there exists an matrix such that for any vector :

Property 3: Composition

The composition of two linear transformations is also a linear transformation. If and are linear transformations, then the composition is a linear transformation.

Property 4: Invertibility

A linear transformation is invertible if there exists a linear transformation such that:

and

Practical Example

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

import numpy as np

# Define a vector
vector = np.array([2, 3])

# Define a transformation matrix (e.g., a scaling matrix)
transformation_matrix = np.array([
    [2, 0],
    [0, 2]
])

# Apply the linear transformation
transformed_vector = np.dot(transformation_matrix, vector)

print("Original Vector:", vector)
print("Transformed Vector:", transformed_vector)

Explanation

  1. We define a vector vector as .
  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.

Output

Original Vector: [2 3]
Transformed Vector: [4 6]

Exercises

Exercise 1

Given the vector and the transformation matrix , apply the linear transformation and find the transformed vector.

Solution

import numpy as np

vector = np.array([1, 2])
transformation_matrix = np.array([
    [3, 0],
    [0, 3]
])

transformed_vector = np.dot(transformation_matrix, vector)

print("Transformed Vector:", transformed_vector)

Output

Transformed Vector: [3 6]

Exercise 2

Prove that the rotation matrix is a linear transformation.

Solution

To prove that is a linear transformation, we need to show that it satisfies additivity and homogeneity.

  1. Additivity:

    Let and .

  2. Homogeneity:

    Let be a scalar and .

Thus, satisfies both additivity and homogeneity, proving it is a linear transformation.

Conclusion

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.

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