Linear Transformations and Matrices

Welcome, students, to a lesson about how algebra can describe movement and change in the plane ✨. Imagine a digital image that gets stretched, flipped, or rotated. The rules behind that change can be written as a linear transformation, and those rules can also be stored in a matrix. In AP Precalculus, this topic connects algebra, geometry, and functions in a powerful way.

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

explain the main ideas and vocabulary of linear transformations and matrices,
use matrices to represent transformations of vectors in the plane,
apply reasoning to predict what happens to shapes after a transformation,
connect transformations to the bigger AP Precalculus topic of functions involving parameters, vectors, and matrices.

This lesson is not directly assessed on the AP Exam, but it strengthens the way you think about functions and vector behavior. Think of it as a tool that helps you see algebra in action 🚀.

What is a linear transformation?

A transformation is a rule that takes an input and produces an output. In this topic, the input is usually a vector like $\begin{pmatrix}x\y\end{pmatrix}$, and the output is another vector.

A linear transformation is a special kind of transformation that preserves two important features:

addition of vectors,
scaling of vectors.

That means if $T$ is a linear transformation, then for vectors $\mathbf{u}$ and $\mathbf{v}$ and scalar $c$, the following rules hold:

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

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

These rules are what make linear transformations predictable and useful.

A simple example is a transformation that stretches every vector horizontally by a factor of $2$. If $T\left(\begin{pmatrix}x\y\end{pmatrix}\right)=\begin{pmatrix}2x\y\end{pmatrix}$, then the $x$-coordinate doubles while the $y$-coordinate stays the same. This is a linear transformation because it follows the rules above.

A transformation like $T\left(\begin{pmatrix}x\y\end{pmatrix}\right)=\begin{pmatrix}x+3\y\end{pmatrix}$ is not linear, because it shifts every point to the right by $3$. That kind of shift is called a translation, and translations do not preserve the origin. Linear transformations always send the zero vector $\begin{pmatrix}0\\0\end{pmatrix}$ to itself.

Matrices as transformation rules

A matrix is a rectangular array of numbers. In this topic, matrices are used to describe linear transformations in a compact way.

For a $2\times 2$ matrix

$$A=\begin{pmatrix}a&b\c&d\end{pmatrix},$$

the transformation of a vector $\begin{pmatrix}x\y\end{pmatrix}$ is found by matrix multiplication:

$$A\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}.$$

This formula tells us exactly how the output depends on the input. Every entry in the matrix has a job.

A very important idea is that the columns of the matrix show what happens to the standard basis vectors:

$$\mathbf{e}_1=\begin{pmatrix}1\\0\end{pmatrix},\qquad \mathbf{e}_2=\begin{pmatrix}0\\1\end{pmatrix}.$$

$$A=\begin{pmatrix}a&b\c&d\end{pmatrix},$$

then

$$A\mathbf{e}_1=\begin{pmatrix}a\c\end{pmatrix},\qquad A\mathbf{e}_2=\begin{pmatrix}b\d\end{pmatrix}.$$

So the first column tells where $\mathbf{e}_1$ goes, and the second column tells where $\mathbf{e}_2$ goes. Once you know those two images, you know the whole transformation.

For example, if

$$A=\begin{pmatrix}2&1\\0&3\end{pmatrix},$$

then

$$A\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}2\\0\end{pmatrix}$$

and

$$A\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}1\\3\end{pmatrix}.$$

That means the unit square is transformed into a new shape whose sides are determined by those two column vectors. This is a big geometric idea: matrices do not just compute answers, they describe movement in the plane 📐.

Common types of linear transformations

Some linear transformations appear often because they have clear geometric effects.

Scaling

A scaling changes size. For example,

$$T\left(\begin{pmatrix}x\y\end{pmatrix}\right)=\begin{pmatrix}3x\\2y\end{pmatrix}$$

stretches horizontally by a factor of $3$ and vertically by a factor of $2$.

Its matrix is

$$\begin{pmatrix}3&0\\0&2\end{pmatrix}.$$

Reflection

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

$$T\left(\begin{pmatrix}x\y\end{pmatrix}\right)=\begin{pmatrix}x\\-y\end{pmatrix},$$

with matrix

$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

Reflection across the $y$-axis is

$$\begin{pmatrix}-1&0\\0&1\end{pmatrix}.$$

Rotation

A rotation turns vectors around the origin. A counterclockwise rotation by angle $\theta$ is represented by

$$\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}.$$

For example, a $90^\circ$ counterclockwise rotation uses

$$\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$$

So the vector $\begin{pmatrix}2\\5\end{pmatrix}$ becomes

$$\begin{pmatrix}-5\\2\end{pmatrix}.$$

Shear

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

$$\begin{pmatrix}1&k\\0&1\end{pmatrix},$$

which sends

$$\begin{pmatrix}x\y\end{pmatrix}$$

$$\begin{pmatrix}x+ky\y\end{pmatrix}.$$

This means the $x$-coordinate depends partly on the $y$-coordinate. Shears are common in graphics and design.

How to apply a matrix to a vector

To use a matrix transformation, multiply the matrix by the vector.

Suppose

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

and

$$\mathbf{v}=\begin{pmatrix}3\\2\end{pmatrix}.$$

Then

$$A\mathbf{v}=\begin{pmatrix}4(3)+(-1)(2)\\2(3)+3(2)\end{pmatrix}=\begin{pmatrix}10\\12\end{pmatrix}.$$

That result means the original vector points somewhere in the plane, and after the transformation it lands at $\begin{pmatrix}10\\12\end{pmatrix}$.

A helpful strategy is to think column by column:

$$A\begin{pmatrix}x\y\end{pmatrix}=x\begin{pmatrix}a\c\end{pmatrix}+y\begin{pmatrix}b\d\end{pmatrix}.$$

This shows that the output is a combination of the matrix columns. That idea connects matrices to vectors and to the broader AP Precalculus theme of combining quantities with parameters.

For instance, if a model uses a parameter $t$ and a vector changes according to

$$\mathbf{v}(t)=\begin{pmatrix}2t\t+1\end{pmatrix},$$

then applying a matrix transformation gives a new vector-valued function:

$$A\mathbf{v}(t).$$

This is one reason matrices matter in functions involving parameters: they let us transform an entire family of inputs at once.

Seeing transformations in the coordinate plane

Linear transformations can be understood visually by tracking a few key points and shapes.

If a transformation sends the unit square to a parallelogram, the columns of the matrix are the sides of that new parallelogram. That makes matrices a bridge between algebraic rules and geometric pictures.

Here is a practical example. Let

$$A=\begin{pmatrix}2&1\\1&2\end{pmatrix}.$$

Then

$$A\mathbf{e}_1=\begin{pmatrix}2\\1\end{pmatrix}$$

and

$$A\mathbf{e}_2=\begin{pmatrix}1\\2\end{pmatrix}.$$

The unit square becomes a parallelogram with those two vectors as sides. Because the columns are not perpendicular and not the same length, the shape is both stretched and tilted.

Another important fact: linear transformations send lines through the origin to lines through the origin. If a set of points lies on a line defined by scalar multiples of one vector, the image will still lie on a line.

Why this matters in AP Precalculus

students, linear transformations and matrices fit into AP Precalculus because they extend the idea of a function. A function maps inputs to outputs, and a matrix transformation does the same thing for vectors.

This topic also connects to the broader ideas of parameters and modeling. For example, if a matrix depends on a parameter $p$,

$$A(p)=\begin{pmatrix}p&0\\0&2\end{pmatrix},$$

then changing $p$ changes the transformation. When $p=1$, the transformation leaves the $x$-direction unchanged; when $p=4$, the $x$-direction is stretched more. This kind of reasoning helps you compare models and predict how outputs change when inputs change.

Matrices also prepare you for more advanced mathematics and science. They are used in computer graphics, engineering, navigation, data analysis, and physics. A game character’s movement, a satellite’s position, or a robot arm’s motion can all involve transformations described by matrices 🤖.

Conclusion

Linear transformations are rules that preserve vector addition and scalar multiplication. Matrices provide a structured way to represent those rules. By looking at the columns of a matrix, you can predict how the transformation affects basis vectors, shapes, and entire graphs in the coordinate plane.

In AP Precalculus, this topic strengthens your understanding of functions, vectors, and parameter-based reasoning. Even though it is not directly assessed on the AP Exam, it builds important mathematical thinking: identify a rule, represent it clearly, and use it to predict outcomes. That is a valuable skill in algebra, geometry, and many real-world applications.

Study Notes

A linear transformation preserves addition and scalar multiplication.
A linear transformation must send $\begin{pmatrix}0\\0\end{pmatrix}$ to itself.
A matrix can represent a linear transformation of vectors in the plane.
For $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$, the output of $\begin{pmatrix}x\y\end{pmatrix}$ is $\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$.
The first column of a matrix shows where $\begin{pmatrix}1\\0\end{pmatrix}$ goes, and the second column shows where $\begin{pmatrix}0\\1\end{pmatrix}$ goes.
Common linear transformations include scaling, reflection, rotation, and shear.
A rotation by angle $\theta$ has matrix $\begin{pmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{pmatrix}$.
Linear transformations connect to functions because both map inputs to outputs.
Parameters in a matrix can change the transformation rule.
This lesson supports the broader AP Precalculus study of functions involving parameters, vectors, and matrices.