How To Prove A Transformation Is Linear

How to Prove a Transformation is Linear: A Comprehensive Guide

Meta Description: Learn how to definitively prove a linear transformation. This guide covers the two crucial properties – additivity and homogeneity – with clear examples and step-by-step explanations. Master linear algebra concepts effortlessly!

Linear transformations are fundamental in linear algebra, forming the bedrock of many advanced concepts. Understanding how to prove a transformation is linear is crucial for success in this field. This guide provides a comprehensive approach, breaking down the process into manageable steps and illustrating with clear examples.

What is a Linear Transformation?

A linear transformation, also known as a linear map, is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This preservation is defined by two key properties:

• Additivity: T(u + v) = T(u) + T(v) for all vectors u and v in the domain.
• Homogeneity: T(cv) = cT(v) for all vectors v in the domain and all scalars c.

To prove a transformation is linear, you must demonstrate that it satisfies both additivity and homogeneity. Failure to satisfy even one property means the transformation is not linear.

Step-by-Step Guide to Proving Linearity

Let's outline the steps involved in proving a transformation is linear:

1. Identify the Transformation: Clearly define the transformation T. This often involves a formula or a set of rules that maps vectors from one vector space to another. For example, T: R² → R² might be defined as T(x, y) = (x + y, 2x).

2. Verify Additivity: Take two arbitrary vectors, u and v, from the domain. Apply the transformation to their sum, T(u + v). Then, separately apply the transformation to each vector and add the results: T(u) + T(v). If T(u + v) = T(u) + T(v), the additivity property holds.

3. Verify Homogeneity: Choose an arbitrary vector v from the domain and an arbitrary scalar c. Apply the transformation to the scalar multiple, T(cv). Then, apply the transformation to v and multiply the result by the scalar: cT(v). If T(cv) = cT(v), the homogeneity property holds.

4. State the Conclusion: If both additivity and homogeneity are verified, conclude that the transformation T is linear. Otherwise, state that the transformation is not linear and provide a counterexample (a specific case where the property fails).

Example: Proving Linearity

Let's consider the transformation T: R² → R² defined by T(x, y) = (2x + y, x - y). Let's prove its linearity:

1. Additivity:

Let u = (x₁, y₁) and v = (x₂, y₂).

T(u + v) = T(x₁ + x₂, y₁ + y₂) = (2(x₁ + x₂) + (y₁ + y₂), (x₁ + x₂) - (y₁ + y₂)) = (2x₁ + 2x₂ + y₁ + y₂, x₁ + x₂ - y₁ - y₂)

T(u) + T(v) = (2x₁ + y₁, x₁ - y₁) + (2x₂ + y₂, x₂ - y₂) = (2x₁ + y₁ + 2x₂ + y₂, x₁ - y₁ + x₂ - y₂) = (2x₁ + 2x₂ + y₁ + y₂, x₁ + x₂ - y₁ - y₂)

Since T(u + v) = T(u) + T(v), the additivity property holds.

2. Homogeneity:

Let v = (x, y) and c be a scalar.

T(cv) = T(cx, cy) = (2(cx) + cy, cx - cy) = (2cx + cy, cx - cy)

cT(v) = c(2x + y, x - y) = (2cx + cy, cx - cy)

Since T(cv) = cT(v), the homogeneity property holds.

3. Conclusion:

Because both additivity and homogeneity are satisfied, the transformation T(x, y) = (2x + y, x - y) is linear.

Common Mistakes to Avoid

• Assuming Linearity: Always explicitly verify both additivity and homogeneity. Don't assume a transformation is linear based on its appearance.
• Incorrect Vector Operations: Ensure you correctly perform vector addition and scalar multiplication.
• Ignoring the Domain and Codomain: Specify the vector spaces involved in the transformation. The properties must hold for all vectors in the domain.

By following these steps and understanding the underlying concepts, you can confidently prove whether a given transformation is linear or not. Remember practice is key to mastering this important linear algebra concept.