Introduction to Linear Transformations

1. What is a Transformation? A Visual Introduction

In linear algebra, "transformation" is essentially a synonym for "function," but it's used to evoke a sense of movement. A transformation $T$ takes an input vector $\mathbf{v}$ and maps it, or "moves" it, to an output vector $T(\mathbf{v})$.

To understand a transformation's effect on the entire space, we can imagine it moving every point (or vector tip) in the space to its new location simultaneously.

While any function that maps vectors to vectors is a transformation, linear algebra focuses on a special, well-behaved class: linear transformations.

Visually, a transformation is linear if it adheres to two simple rules:

1. All lines remain lines. The transformation can't curve or bend straight lines.
2. The origin remains fixed. The point $(0,0)$ (or its higher-dimensional equivalent) must stay put.

This implies that grid lines must remain parallel and evenly spaced after the transformation.

Common examples of linear transformations that you can visualize include:

• Scaling: Stretching or shrinking the entire space.
• Rotation: Rotating the space around the origin.
• Shearing: Pushing one part of the space in a direction parallel to an axis, like sliding the top of a deck of cards.

2. The Formal Definition of Linearity

The visual intuition is powerful, but we need a precise mathematical definition to work with. A transformation $T$ from a vector space $V$ to a vector space $W$ is linear if it satisfies two properties for all vectors $\mathbf{u},\mathbf{v}$ in $V$ and any scalar $c$:

1. Additivity: $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$

   • In words: The transformation of a sum of two vectors is the same as the sum of their individual transformations. You can either add the vectors first and then transform, or transform them first and then add.

2. Homogeneity (Scalar Multiplication): $T(c\mathbf{v})=cT(\mathbf{v})$

   • In words: The transformation of a scaled vector is the same as scaling the transformed vector. You can either scale first and then transform, or transform first and then scale.

These two algebraic rules are what enforce the geometric picture of "grid lines stay parallel and evenly spaced."

A More Efficient Test

Checking two separate properties can be tedious. Luckily, they can be combined into a single, equivalent condition:

A transformation $T$ is linear if and only if
$T(c\mathbf{u}+d\mathbf{v})=cT(\mathbf{u})+dT(\mathbf{v})$
for all vectors $\mathbf{u},\mathbf{v}$ and all scalars $c,d$. This single check verifies both additivity and homogeneity at once and is often the most practical way to prove a transformation is linear. The video you just saw discusses this combined property right after the initial definition (from 2:52 to 5:23), which you can review if you're interested in the proof of equivalence.

3. How to Determine if a Transformation is Linear

Now let's put this definition to work. How do you test a given function to see if it's a linear transformation?

A Quick Disqualification Test

Before diving into the full definition, there's a quick check you can do. A consequence of the homogeneity property is that a linear transformation must map the zero vector to the zero vector.
Let $c=0$, then $T(0)=T(0\cdot \mathbf{v})=0\cdot T(\mathbf{v})=0$.

So, if $T(0)\ne 0$, the transformation is not linear.

Warning: If $T(0)=0$, it does not automatically mean the transformation is linear. This is a necessary, but not sufficient, condition. It's only useful for quickly ruling out non-linear transformations.

Applying the Full Definition: Examples

Let's look at some concrete examples.

Example 1: Matrix Multiplication
This is the most important class of linear transformations for our purposes. Let $A$ be an $m\times n$ matrix. The transformation $T(\mathbf{x})=A\mathbf{x}$ maps vectors from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$. Is this linear?

Let's check the two properties using facts about matrix algebra:

1. Additivity: $T(\mathbf{u}+\mathbf{v})=A(\mathbf{u}+\mathbf{v})=A\mathbf{u}+A\mathbf{v}=T(\mathbf{u})+T(\mathbf{v})$. (This works because of the distributive property of matrix multiplication).
2. Homogeneity: $T(c\mathbf{u})=A(c\mathbf{u})=c(A\mathbf{u})=cT(\mathbf{u})$. (This works because scalars can be pulled out of matrix-vector products).

Both properties hold, so multiplication by any matrix is always a linear transformation. This is the crucial link we sought: the abstract idea of a "linear transformation" in ${\mathbb{R}}^n$ is embodied by the concrete operation of matrix multiplication.

Example 2: The Derivative Operator
Your background in calculus will be useful here. Let $V$ be the vector space of all differentiable functions. Is the derivative operator $D$, defined by $D(f)=f^{\prime }$, a linear transformation?

1. Additivity: $D(f+g)=(f+g{)}^{\prime }=f^{\prime }+g^{\prime }=D(f)+D(g)$. (The derivative of a sum is the sum of the derivatives).
2. Homogeneity: $D(cf)=(cf{)}^{\prime }=c{f}^{\prime }=cD(f)$. (Constants can be pulled out of derivatives).

Yes, differentiation is a linear transformation! This shows that the concept of linearity is much broader than just matrices and applies to spaces of functions, which is fundamental in areas like quantum mechanics (where operators are linear transformations on function spaces).

Example 3: A Non-Linear Transformation
Consider the transformation $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by $T\left(\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\right)=\left[\begin{array}{c}{x}_1^2\\ x_2\end{array}\right]$.

Let's test it. First, the quick check: $T(0)=T\left(\left[\begin{array}{c}0\\ 0\end{array}\right]\right)=\left[\begin{array}{c}{0}^2\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$. The test doesn't rule it out, so we must use the full definition.

Let's check homogeneity. Let $\mathbf{v}=\left[\begin{array}{c}1\\ 1\end{array}\right]$ and $c=2$.

• $T(c\mathbf{v})=T\left(2\left[\begin{array}{c}1\\ 1\end{array}\right]\right)=T\left(\left[\begin{array}{c}2\\ 2\end{array}\right]\right)=\left[\begin{array}{c}{2}^2\\ 2\end{array}\right]=\left[\begin{array}{c}4\\ 2\end{array}\right]$.
• $cT(\mathbf{v})=2\cdot T\left(\left[\begin{array}{c}1\\ 1\end{array}\right]\right)=2\cdot \left[\begin{array}{c}{1}^2\\ 1\end{array}\right]=2\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}2\\ 2\end{array}\right]$.

Since $T(c\mathbf{v})\ne cT(\mathbf{v})$, the transformation is not linear. We only need to find one counterexample to disprove linearity.

As a rule of thumb, if the components of the output vector involve squares, products of components, or other non-linear functions (like $\sin (x_1)$ or $|x_1|$), the transformation is likely not linear.

Conclusion

In this lesson, we introduced the fundamental concept of a linear transformation, moving from a visual, intuitive understanding to a formal algebraic definition.

Key Takeaways:

• A transformation is a function that maps vectors to vectors, visualized as a "movement" of space.
• A transformation is linear if it keeps grid lines straight, parallel, and evenly spaced, and leaves the origin fixed.
• The formal definition of linearity requires two properties: additivity ($T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$) and homogeneity ($T(c\mathbf{v})=cT(\mathbf{v})$).
• A quick way to spot a non-linear transformation is to check if $T(0)\ne 0$.
• Matrix multiplication is a linear transformation. This is the core example that we will build upon.

Preview of the Next Lesson:
We have seen that every matrix gives rise to a linear transformation. The natural next question is: does it work the other way around? For any linear transformation between spaces like ${\mathbb{R}}^n$ and ${\mathbb{R}}^m$, can we find a matrix that represents it? The answer is a resounding yes! In the next lesson, we will learn how to construct the matrix representation for any given linear transformation. This will solidify the powerful equivalence between the geometry of linear transformations and the algebra of matrices.