Definition of a Linear Transformation

students, when you map one set of vectors to another, you are often looking at a transformation. In linear algebra, some transformations are especially important because they preserve the structure of vectors in a predictable way. These are called linear transformations ✨. Understanding this idea is a major step in learning how matrices, geometry, and vector spaces all fit together.

In this lesson, you will learn what a linear transformation is, why it matters, and how to tell whether a transformation is linear. By the end, you should be able to explain the definition clearly, test examples, and connect the idea to the wider topic of linear transformations.

What Is a Transformation?

A transformation is a rule that takes each vector in one vector space and sends it to a vector in another vector space, or possibly the same one. You can think of it like a machine that processes input vectors and produces output vectors 🚀.

For example, if a transformation takes a vector $\begin{bmatrix}x\y\end{bmatrix}$ and sends it to $\begin{bmatrix}x+2\y\end{bmatrix}$, then every input vector gets moved to a new location.

A transformation is often written as $T$. So if $T$ is a transformation and $\mathbf{v}$ is a vector, we write the output as $T(\mathbf{v})$.

The important question is not just whether a rule sends vectors somewhere. The real question is whether it preserves the structure of vector addition and scalar multiplication. That is what makes a transformation linear.

The Definition of a Linear Transformation

A transformation $T$ is called linear if it satisfies both of these properties for all vectors $\mathbf{u}$ and $\mathbf{v}$ and all scalars $c$:

$$T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$$

$$T(c\mathbf{u})=cT(\mathbf{u})$$

These two rules are the heart of the definition.

The first rule says that if you add two vectors first and then transform the result, you get the same answer as transforming each vector first and then adding the outputs. This is called additivity.

The second rule says that if you scale a vector first and then transform it, you get the same answer as transforming first and then scaling. This is called homogeneity or scalar multiplication preservation.

Together, these properties mean that the transformation respects the algebra of vectors. Linear transformations do not bend, curve, or shift the space in ways that destroy this structure.

A Useful Shortcut: Testing Linearity

To check whether a transformation is linear, students, you usually test the two defining properties directly.

Here is a simple process:

1. Pick general vectors $\mathbf{u}$ and $\mathbf{v}$.
2. Compute $T(\mathbf{u}+\mathbf{v})$.
3. Compute $T(\mathbf{u})+T(\mathbf{v})$.
4. Compare the two results.
5. Then check whether $T(c\mathbf{u})=cT(\mathbf{u})$.

If both properties always hold, the transformation is linear.

There is also an important fact: if a transformation is linear, then it must send the zero vector to the zero vector. That is,

$$T(\mathbf{0})=\mathbf{0}$$

Why? Because if $T$ is linear, then $T(\mathbf{0})=T(0\mathbf{u})=0T(\mathbf{u})=\mathbf{0}$. This gives a fast way to rule out many non-linear transformations.

For example, any transformation that adds a constant vector, such as $T(\mathbf{x})=\mathbf{x}+\mathbf{b}$ where $\mathbf{b}\neq \mathbf{0}$, is not linear because it sends $\mathbf{0}$ to $\mathbf{b}$, not to $\mathbf{0}$.

Examples of Linear Transformations

Let’s look at some examples that are linear.

Example 1: Scaling

Define $T:\mathbb{R}^2\to\mathbb{R}^2$ by

$$T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)=\begin{bmatrix}2x\\2y\end{bmatrix}$$

This is linear because multiplying every input vector by $2$ preserves both addition and scalar multiplication.

Check additivity:

$$T\!\left(\begin{bmatrix}x_1\y_1\end{bmatrix}+\begin{bmatrix}x_2\y_2\end{bmatrix}\right)=T\!\left(\begin{bmatrix}x_1+x_2\y_1+y_2\end{bmatrix}\right)=\begin{bmatrix}2(x_1+x_2)\\2(y_1+y_2)\end{bmatrix}$$

and

$$T\!\left(\begin{bmatrix}x_1\y_1\end{bmatrix}\right)+T\!\left(\begin{bmatrix}x_2\y_2\end{bmatrix}\right)=\begin{bmatrix}2x_1\\2y_1\end{bmatrix}+\begin{bmatrix}2x_2\\2y_2\end{bmatrix}=\begin{bmatrix}2(x_1+x_2)\\2(y_1+y_2)\end{bmatrix}$$

These match ✅.

Example 2: Rotation

A rotation around the origin in the plane is also linear. If $T$ rotates every vector by a fixed angle, then vector addition and scalar multiplication are preserved.

This happens because the transformation changes direction but keeps the origin fixed and respects how vectors combine. Rotations are common in computer graphics, robotics, and engineering.

Example 3: Matrix Transformation

If $A$ is a matrix, then the rule

$$T(\mathbf{x})=A\mathbf{x}$$

is always linear.

For instance, if

$$A=\begin{bmatrix}1&3\\0&-2\end{bmatrix}$$

then

$$T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)=\begin{bmatrix}x+3y\\-2y\end{bmatrix}$$

This is linear because matrix multiplication automatically preserves addition and scalar multiplication.

This fact is one of the most important connections in linear algebra: every matrix defines a linear transformation, and every linear transformation between finite-dimensional vector spaces can be represented by a matrix once bases are chosen.

Examples of Nonlinear Transformations

Not every transformation is linear. In fact, many familiar rules are not linear.

Example 1: Translation

Consider

$$T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)=\begin{bmatrix}x+1\y\end{bmatrix}$$

This is not linear because

$$T\!\left(\begin{bmatrix}0\\0\end{bmatrix}\right)=\begin{bmatrix}1\\0\end{bmatrix}$$

but a linear transformation must send $\mathbf{0}$ to $\mathbf{0}$.

Example 2: Squaring a Coordinate

Consider

$$T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)=\begin{bmatrix}x^2\y\end{bmatrix}$$

This is not linear because scalar multiplication does not work properly. If $c=2$, then

$$T\!\left(2\begin{bmatrix}x\y\end{bmatrix}\right)=\begin{bmatrix}(2x)^2\\2y\end{bmatrix}=\begin{bmatrix}4x^2\\2y\end{bmatrix}$$

but

$$2T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)=2\begin{bmatrix}x^2\y\end{bmatrix}=\begin{bmatrix}2x^2\\2y\end{bmatrix}$$

These are not the same unless $x=0$.

Why Linearity Matters

Linear transformations are useful because they are predictable and easy to analyze. They show up in many real situations 📘.

For example:

• In computer graphics, linear transformations help rotate, stretch, and reflect images.
• In engineering, they model systems where outputs depend proportionally on inputs.
• In data science, matrix transformations are used to process and represent data.
• In physics, linear approximations help describe small changes around equilibrium.

The reason linear transformations are so important is that they let us use algebra to study geometry. Instead of drawing every possible input-output pair, we can often understand the entire transformation from a few key properties.

Another key idea is that a linear transformation is fully determined by what it does to a basis. In $\mathbb{R}^n$, if you know how a linear transformation acts on the standard basis vectors, you can determine it everywhere.

For example, if $T:\mathbb{R}^2\to\mathbb{R}^2$ is linear, then knowing $T\!\left(\begin{bmatrix}1\\0\end{bmatrix}\right)$ and $T\!\left(\begin{bmatrix}0\\1\end{bmatrix}\right)$ is enough to find $T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)$ for any vector $\begin{bmatrix}x\y\end{bmatrix}$.

This works because

$$\begin{bmatrix}x\y\end{bmatrix}=x\begin{bmatrix}1\\0\end{bmatrix}+y\begin{bmatrix}0\\1\end{bmatrix}$$

and linearity gives

$$T\!\left(\begin{bmatrix}x\y\end{bmatrix}\right)=xT\!\left(\begin{bmatrix}1\\0\end{bmatrix}\right)+yT\!\left(\begin{bmatrix}0\\1\end{bmatrix}\right)$$

That formula is a powerful tool.

A Real-World Analogy

Imagine a vending machine that follows exact pricing rules. If buying two identical snacks costs exactly twice as much as buying one, and if buying $c$ of something costs $c$ times the single price, then the system is behaving linearly in that sense.

A linear transformation behaves like that kind of rule for vectors: combine inputs first or combine outputs later, and the result is the same. But if the machine adds a fixed fee every time, the rule is no longer linear. That fixed fee is like a translation term, and it breaks linearity.

Conclusion

students, the definition of a linear transformation is simple but powerful: it is a transformation that preserves vector addition and scalar multiplication. In formula form, $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$.

This idea is central to linear algebra because it connects vectors, matrices, geometry, and applications. Once you understand linear transformations, you are ready to study matrices more deeply and solve many problems in a structured way.

Study Notes

• A transformation is a rule that sends vectors to vectors.
• A transformation $T$ is linear if it satisfies both $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$.
• Linear transformations preserve the structure of vector addition and scalar multiplication.
• Every linear transformation sends the zero vector to the zero vector, so $T(\mathbf{0})=\mathbf{0}$.
• Matrix multiplication defines a linear transformation: $T(\mathbf{x})=A\mathbf{x}$.
• Rotations, reflections, and scalings about the origin are linear.
• Translations are not linear because they do not send $\mathbf{0}$ to $\mathbf{0}$.
• A nonlinear rule often includes constants, powers like $x^2$, or other operations that break the two defining properties.
• Knowing the image of a basis is enough to determine a linear transformation.
• Linear transformations are important in geometry, computer graphics, science, and engineering.