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Formalizing Linear Transformations

In the previous sections, we introduced the concept of linear transformations and how they can be represented by matrices. We explored everything in a visual manner, using 2D and 3D space to help us understand the transformations.

In this section, we will formalize the concept of linear transformations.

Table of Contents​

Functions and Sets​

Before we dive into linear transformations, let's first review some basic concepts from functions and sets.

A function is a rule that assigns each element in a set to exactly one element in another set. For example, the function assigns each real number to the real number .

We will need to understand some terminology related to functions:

  • Every function has a domain and a image (or range), representing the set of possible inputs and possible outputs, respectively.
  • The image lives in the codomain, which is the set in which the function outputs are contained.

We can draw a diagram to represent visually how a function maps elements from the domain to the range:

For example, consider the function :

  • The domain is all real numbers, .
  • The codomain is also all real numbers, .
  • The range is all non-negative real numbers, .

We can notate the function as , which reads as " maps from the real numbers to the non-negative real numbers".

What Linear Transformations Really Are​

Now, let's dive into linear transformations. Linear transformations are a special kind of function; previously, we stated that they act on vectors. For example, a linear transformation might take a 2D vector and output another 2D vector. Then, we can write the transformation as .

Recall that when we apply a linear transformation to a vector, if we express the vector as a linear combination of the basis vectors, the transformation can be applied to each basis vector separately. Then, the coefficients of the linear combination are preserved.

For example, consider a linear transformation and a vector . Then, . This is due to the linearity of the transformation, which we said means that the transformation:

  • Preserves the origin.
  • Preserves lines, which stay parallel and evenly spaced.

Let's see how we can formalize this. We start with a premise:

We will go from this to the formal definition of a linear transformation.

Consider applying the transformation to the sum of two vectors and , where and . Then:

Additionally, consider applying the transformation to a scalar multiple of a vector, , where :

So we have shown that a linear transformation satisfies the following properties:

  1. . This is called the additivity of the transformation.
  2. . This is called the homogeneity of the transformation.

These properties are the formal definition of what it means for a transformation to be linear.

We can construct a formal definition of a linear transformation from these properties:

The transformation is linear if it satisfies the following properties:

  1. Additivity: for all vectors .
  2. Homogeneity: for all vectors and scalars .

A good exercise is to consider our geometric understanding of linear transformations and see how it fits with this formal definition without getting lost in the notation. Are the properties of additivity and homogeneity consistent with our understanding of how linear transformations behave geometrically (preserving the origin and lines)?

Example Problem: Verifying Linearity of Composite Transformations​

Let and be two linear transformations. Prove that the composite transformation is also a linear transformation.

Before proving why it is true, let's first consider why it should be true. A vector , as previously mentioned, can be expressed as a linear combination of the basis vectors. When we apply the transformation to , it changes the basis vectors, but keeps the coefficients the same. Then, when we apply the transformation to the result, it changes the basis vectors again, but keeps the coefficients the same.

This means that overall, the composite transformation is equivalent to changing the basis vectors and keeping the coefficients the same, which is what a linear transformation does. An example might help:

  • Let and , where scales the -axis by 2 and rotates the plane by 90 degrees.
  • Then, and , meaning .
  • Similarly, and , meaning .
  • This is equivalent to a transformation that transforms the basis vectors to and ; .

Now, let's prove this formally.

Proof: To prove that is a linear transformation, we need to show that it satisfies the properties of additivity and homogeneity.

  1. Additivity: Let . Then:

    Hence, satisfies the additivity property.

  2. Homogeneity: Let and . Then:

    Hence, satisfies the homogeneity property.

Therefore, is a linear transformation.

Linear Transformations of Different Sets​