Transformations of the Plane

students, in this lesson you will explore how points and shapes in the plane can move, stretch, flip, and rotate using linear transformations. These ideas are a foundation of linear algebra and show up in computer graphics, engineering, physics, and data science 🎯

What You Will Learn

By the end of this lesson, you should be able to:

explain the main ideas and vocabulary of transformations of the plane,
apply linear algebra reasoning to describe what a transformation does,
connect plane transformations to the broader topic of linear transformations,
summarize why these transformations matter in linear algebra,
use examples to identify and analyze transformations.

A transformation is a rule that takes each point in the plane and sends it to a new point. Think of drawing a shape on a sheet of transparent plastic and then moving the sheet. The picture changes position, and sometimes its size or orientation changes too ✨

What Is a Transformation?

A transformation of the plane is a function that takes every point $(x, y)$ and maps it to another point $(x', y')$. In linear algebra, we often study transformations as functions from $\mathbb{R}^2$ to $\mathbb{R}^2$.

For example, a transformation might send a point according to the rule

$$T(x, y) = (2x, y)$$

This means the $x$-coordinate is doubled, while the $y$-coordinate stays the same. A point like $(3, 4)$ would move to $(6, 4)$. This kind of transformation stretches the plane horizontally.

Not every transformation is linear. To be a linear transformation, a rule must satisfy two important properties:

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

and

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

for any vectors $\mathbf{u}$, $\mathbf{v}$, and scalar $c$.

These rules say that the transformation respects vector addition and scalar multiplication. That makes linear transformations especially useful because they preserve the structure of the plane.

Common Transformations of the Plane

Several basic transformations appear again and again in linear algebra.

1. Stretching and compressing

A stretching transformation changes the size of figures in one direction. For instance,

$$T(x, y) = (3x, y)$$

stretches points horizontally by a factor of $3$. A square would become a rectangle. If the rule were

$$T(x, y) = (x, \tfrac{1}{2}y)$$

then the plane would be compressed vertically by a factor of $\tfrac{1}{2}$.

These transformations are linear because they can be written using matrices, such as

$$\begin{pmatrix}3 & 0\\0 & 1\end{pmatrix}$$

for the horizontal stretch.

2. Reflections

A reflection flips the plane over a line. Reflection across the $x$-axis is given by

$$T(x, y) = (x, -y)$$

so the point $(2, 5)$ becomes $(2, -5)$. Reflection across the $y$-axis is

$$T(x, y) = (-x, y)$$

which sends $(2, 5)$ to $(-2, 5)$.

Reflections are linear because they preserve addition and scalar multiplication. They also reverse orientation, which means a shape that was “right-facing” can become “left-facing.” This is easy to see in a mirror 🪞

3. Rotations

A rotation turns every point around the origin by a fixed angle. A counterclockwise rotation by $90^\circ$ is

$$T(x, y) = (-y, x)$$

So $(3, 1)$ becomes $(-1, 3)$. Rotation by $180^\circ$ is

$$T(x, y) = (-x, -y)$$

Rotations are linear transformations because they preserve the origin and the vector-space structure, even though they change direction.

4. Shears

A shear slants a shape without necessarily changing its area in the same way as stretching. A horizontal shear can be written as

$$T(x, y) = (x + ky, y)$$

where $k$ is a constant. If $k = 2$, then

$$T(x, y) = (x + 2y, y)$$

A point like $(1, 3)$ becomes $(7, 3)$. Shears are linear because they can be represented by matrices and they preserve lines, although not necessarily angles.

Matrix Representation of Transformations

One of the biggest ideas in linear algebra is that every linear transformation of $\mathbb{R}^2$ can be represented by a $2 \times 2$ matrix.

$$T(x, y) = (ax + by, cx + dy)$$

then the transformation can be written as

$$T\left(\begin{pmatrix}x\y\end{pmatrix}\right) = \begin{pmatrix}a & b\c & d\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}$$

This matrix tells us exactly how the transformation acts on every vector in the plane.

A powerful idea is that a matrix is determined by what it does to the basis vectors

$$\mathbf{e}_1 = \begin{pmatrix}1\\0\end{pmatrix} \quad \text{and} \quad \mathbf{e}_2 = \begin{pmatrix}0\\1\end{pmatrix}$$

If we know $T(\mathbf{e}_1)$ and $T(\mathbf{e}_2)$, then we know the whole transformation. For example, if

$$T\left(\begin{pmatrix}1\\0\end{pmatrix}\right) = \begin{pmatrix}3\\1\end{pmatrix} \quad \text{and} \quad T\left(\begin{pmatrix}0\\1\end{pmatrix}\right) = \begin{pmatrix}2\\4\end{pmatrix}$$

then the matrix of $T$ is

$$\begin{pmatrix}3 & 2\\1 & 4\end{pmatrix}$$

This works because any vector $\begin{pmatrix}x\y\end{pmatrix}$ can be written as

$$x\mathbf{e}_1 + y\mathbf{e}_2$$

and linearity tells us how $T$ acts on sums and scalar multiples.

Examples and Real-World Connections

Suppose a game designer wants to move a character’s sprite on a screen. If the designer rotates the sprite, the shape is transformed in the plane. If the designer stretches it, the character might look taller or wider. These changes are built using the same kinds of transformations studied in linear algebra 🎮

Another example is map-making. A map projection converts points on the Earth’s surface to points on a flat page. While many map projections are not linear, they still use the idea of transforming one plane-like surface into another. In computer graphics, transformations are combined to animate objects, zoom in, or mirror images.

Let’s look at a simple example. Consider the triangle with vertices $(0,0)$, $(1,0)$, and $(0,2)$. Under the transformation

$$T(x, y) = (x + y, y)$$

the vertices become:

$(0,0) \mapsto (0,0)$
$(1,0) \mapsto (1,0)$
$(0,2) \mapsto (2,2)$

The triangle is sheared to the right. The shape is still a triangle, but it looks slanted. This shows how a transformation can change appearance while keeping lines as lines.

Why Linear Transformations Matter

Linear transformations are important because they are predictable and structured. They always send the zero vector to the zero vector:

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

They also preserve the way vectors combine. This makes them much easier to study than arbitrary functions.

In linear algebra, transformations of the plane help connect geometric ideas with algebraic tools. A picture of a rotated square becomes a matrix. A matrix multiplication becomes a changed shape. This connection is one of the central achievements of linear algebra.

It is also useful to notice what linear transformations do not do. They do not bend the plane into a curve. They do not move the origin to a different point. A transformation like

$$T(x, y) = (x + 1, y)$$

is a translation, but it is not linear because

$$T\left(\begin{pmatrix}0\\0\end{pmatrix}\right) = \begin{pmatrix}1\\0\end{pmatrix} \neq \begin{pmatrix}0\\0\end{pmatrix}$$

So translations are transformations of the plane, but not linear transformations.

Conclusion

students, transformations of the plane describe rules that move points in $\mathbb{R}^2$ to new locations. Some transformations stretch, reflect, rotate, or shear. In linear algebra, the most important of these are linear transformations, because they preserve vector addition and scalar multiplication. Every linear transformation of the plane can be represented by a $2 \times 2$ matrix, making geometry and algebra work together. Understanding these transformations gives you a strong foundation for later topics in linear algebra and for many applications in the real world.

Study Notes

A transformation of the plane is a rule that sends each point $(x, y)$ to another point $(x', y')$.
Linear transformations satisfy $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$.
Common plane transformations include stretches, reflections, rotations, and shears.
A linear transformation of $\mathbb{R}^2$ can be written using a $2 \times 2$ matrix.
The matrix of a transformation is determined by the images of $\mathbf{e}_1=\begin{pmatrix}1\\0\end{pmatrix}$ and $\mathbf{e}_2=\begin{pmatrix}0\\1\end{pmatrix}$.
Rotations and reflections about the origin are linear transformations.
A translation like $T(x, y)=(x+1, y)$ is a transformation of the plane, but not a linear transformation.
Linear transformations preserve the origin, straight lines, and vector-space structure.
Plane transformations connect geometry, matrices, and real-world uses like graphics and design.