Verifying Transformations Using Input-Output Pairs

Introduction

Key Concepts

Understanding Linear Transformations

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

• Additivity: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
• Homogeneity: \( T(c\mathbf{u}) = cT(\mathbf{u}) \)

These properties ensure that the structure of the vector space is maintained under the transformation.

Input-Output Pairs in Linear Transformations

An input-output pair consists of an input vector \( \mathbf{x} \) and its corresponding output vector \( T(\mathbf{x}) \) under the transformation \( T \). These pairs are instrumental in verifying whether a given function is a linear transformation.

For example, consider the transformation \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) defined by: $$ T\left(\begin{bmatrix} x \\ y \end{bmatrix}\right) = \begin{bmatrix} 2x + y \\ x - y \end{bmatrix} $$ An input-output pair for this transformation could be: $$ \mathbf{x} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad T(\mathbf{x}) = \begin{bmatrix} 4 \\ -1 \end{bmatrix} $$

Verifying Transformations Using Input-Output Pairs

To verify that a function \( T \) is a linear transformation using input-output pairs, follow these steps:

1. Select Inputs: Choose multiple input vectors \( \mathbf{u}, \mathbf{v} \), and scalars \( c \).
2. Apply Transformation: Compute the outputs \( T(\mathbf{u}), T(\mathbf{v}), T(c\mathbf{u}) \), and \( T(\mathbf{u} + \mathbf{v}) \).
3. Check Additivity: Verify that \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
4. Check Homogeneity: Verify that \( T(c\mathbf{u}) = cT(\mathbf{u}) \).

If both properties hold for all chosen pairs and scalars, \( T \) is a linear transformation.

Examples and Applications

Consider the transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by: $$ T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} x + 2y \\ 3y \\ z \end{bmatrix} $$ To verify linearity, take input vectors \( \mathbf{u} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \), and scalar \( c = 2 \). Compute: $$ T(\mathbf{u}) = \begin{bmatrix} 1 + 2(0) \\ 3(0) \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} $$ $$ T(\mathbf{v}) = \begin{bmatrix} 0 + 2(1) \\ 3(1) \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} $$ $$ T(\mathbf{u} + \mathbf{v}) = T\left(\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\right) = \begin{bmatrix} 1 + 2(1) \\ 3(1) \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} $$ $$ T(2\mathbf{u}) = T\left(\begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix}\right) = \begin{bmatrix} 2 + 2(0) \\ 3(0) \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} $$ Now, check additivity and homogeneity: $$ T(\mathbf{u}) + T(\mathbf{v}) = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 0 \end{bmatrix} = T(\mathbf{u} + \mathbf{v}) $$ $$ 2T(\mathbf{u}) = 2\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} = T(2\mathbf{u}) $$> Since both properties hold, \( T \) is a linear transformation.

**Applications of Verifying Linear Transformations:**

• Computer Graphics: Linear transformations are used to manipulate images and models, including scaling, rotation, and translation.
• Engineering: Analyzing stresses and strains in materials involves linear transformations.
• Economics: Modeling economic systems and optimizing resources often employs linear transformation techniques.

Comparison Table

Summary and Key Takeaways