Verifying Linear Transformations: A Step-by-Step Guide

Hey guys! Let's dive into the world of linear transformations. They're super important in linear algebra, and understanding them is key to unlocking many concepts. In this article, we'll go through the process of verifying whether a given function is a linear transformation. Specifically, we will analyze a function T that maps vectors from F³ to F³. This function, defined as T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃), will be our main focus. We'll break down the necessary steps, ensuring you grasp the core principles of linear transformations. So, let's get started and make sure you fully comprehend this critical concept in linear algebra!

Understanding the Basics of Linear Transformations

Alright, before we jump into the specifics of our example, let's lay down the groundwork. Linear transformations are special types of functions that preserve vector addition and scalar multiplication. This basically means they play nicely with these two fundamental operations. If a function T satisfies two conditions, it's considered a linear transformation:

1. Additivity: T(u + v) = T(u) + T(v) for all vectors u and v in the domain. This means that the transformation of the sum of two vectors is the same as the sum of the transformations of each vector individually.
2. Homogeneity: T(cu) = cT(u) for all scalars c and vectors u in the domain. This means that the transformation of a scalar multiple of a vector is the same as the scalar multiple of the transformation of the vector.

These two properties are the essence of linearity. A function must satisfy both to be considered a linear transformation. If either one fails, then the function is not a linear transformation. The preservation of these operations allows us to work with vectors in a way that's consistent and predictable. Linear transformations are used everywhere, from computer graphics to physics and engineering. Understanding what defines them is the first step toward mastering their applications. We are now ready to apply these rules to determine whether a specific function T is a linear transformation.

Verifying Additivity: The First Test

Okay, time to get our hands dirty! Let's test our function T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) for additivity. The goal here is to show that T(u + v) = T(u) + T(v), where u and v are any two vectors in F³.

Let's denote our vectors as follows:

• u = (x₁, x₂, x₃)
• v = (y₁, y₂, y₃)

First, let's find the sum of these two vectors, u + v:

• u + v = (x₁ + y₁, x₂ + y₂, x₃ + y₃)

Now, let's apply the transformation T to the sum u + v:

• T(u + v) = T(x₁ + y₁, x₂ + y₂, x₃ + y₃)

Using the definition of T, we get:

• T(u + v) = ((*x₁ + y₁) - (*x₂ + y₂) + 2(x₃ + y₃), 2(*x₁ + y₁) + (x₂ + y₂), -(*x₁ + y₁) - 2(*x₂ + y₂) + 2(x₃ + y₃))

Simplifying this, we get:

• T(u + v) = (x₁ + y₁ - x₂ - y₂ + 2x₃ + 2y₃, 2x₁ + 2y₁ + x₂ + y₂, -x₁ - y₁ - 2x₂ - 2y₂ + 2x₃ + 2y₃)

Now, let's find T(u) and T(v) separately:

• T(u) = T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃)
• T(v) = T(y₁, y₂, y₃) = (y₁ - y₂ + 2y₃, 2y₁ + y₂, -y₁ - 2y₂ + 2y₃)

Then, we add T(u) and T(v):

• T(u) + T(v) = (x₁ - x₂ + 2x₃ + y₁ - y₂ + 2y₃, 2x₁ + x₂ + 2y₁ + y₂, -x₁ - 2x₂ + 2x₃ - y₁ - 2y₂ + 2y₃)

Comparing T(u + v) and T(u) + T(v), we see that they are equal. Therefore, our function T satisfies the additivity property. That is a fantastic start! But, we're not done yet – we need to check for homogeneity.

Verifying Homogeneity: The Second Test

Alright, let's roll up our sleeves and check if our function T also satisfies the homogeneity property. Remember, we need to show that T(cu) = cT(u) for any scalar c and vector u.

Let's stick with our vector u = (x₁, x₂, x₃) and consider a scalar c. First, we find the scalar multiple of the vector u:

• cu = (cx₁, cx₂, cx₃)

Now, let's apply the transformation T to this scalar multiple:

• T(cu) = T(cx₁, cx₂, cx₃)

Using the definition of T, we get:

• T(cu) = (cx₁ - cx₂ + 2cx₃, 2cx₁ + cx₂, -cx₁ - 2cx₂ + 2cx₃)

We can factor out the scalar c:

• T(cu) = c( x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃)

Now, let's find T(u) and multiply it by c:

• T(u) = T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃)

• cT(u) = c(x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃)

Comparing T(cu) and cT(u), we see that they are equal. So, our function T satisfies the homogeneity property as well. This means that our function T preserves scalar multiplication!

Conclusion: Is T a Linear Transformation?

We've done it! We have verified that our function T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) satisfies both the additivity and homogeneity properties. Since it passes both tests, we can confidently conclude that T is indeed a linear transformation. This means it preserves vector addition and scalar multiplication, which makes it a well-behaved function within the context of linear algebra.

Great job, guys! You now know how to check if a function is a linear transformation! The key is to remember to check both additivity and homogeneity. This skill is fundamental for understanding more complex linear algebra concepts. Keep practicing, and you will get the hang of it in no time!

In summary:

• T is a linear transformation if and only if it satisfies additivity: T(u + v) = T(u) + T(v).
• T is a linear transformation if and only if it satisfies homogeneity: T(cu) = cT(u).

Congratulations on making it through this article! You should now have a strong grasp of what it takes to verify if a function is a linear transformation. Keep up the great work!