Matrix Representations in Linear Transformations
Welcome, students 👋 In this lesson, you will learn how a linear transformation can be described using a matrix. This is one of the most important ideas in linear algebra because it turns a geometric idea into an algebraic tool. Instead of thinking only about arrows moving on a graph, we can use a matrix to calculate where every vector goes.
Lesson Objectives
By the end of this lesson, students, you should be able to:
- explain what a matrix representation of a linear transformation means,
- connect a transformation rule to its matrix,
- use a matrix to find the image of a vector,
- understand how matrix representations fit into the bigger topic of linear transformations,
- use examples to justify why matrices are useful in linear algebra.
Think of this lesson as learning the “code” behind a transformation. A matrix is like a machine’s instruction sheet 📘. If you know the matrix, you can predict how the transformation behaves.
What Is a Matrix Representation?
A linear transformation is a function $T$ that takes vectors as inputs and gives vectors as outputs. In many courses, the input and output vectors live in $\mathbb{R}^n$. The transformation is called linear if it satisfies both of these rules:
$$T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$$
and
$$T(c\mathbf{u})=cT(\mathbf{u})$$
for all vectors $\mathbf{u}$ and $\mathbf{v}$ and all scalars $c$.
A matrix representation is a matrix $A$ such that
$$T(\mathbf{x})=A\mathbf{x}$$
for every vector $\mathbf{x}$ in the domain. This means the matrix and the transformation do the same job. The matrix is the algebraic form of the transformation.
For example, if
$$A=\begin{bmatrix}2 & 0\\ 0 & 3\end{bmatrix},$$
then
$$T\left(\begin{bmatrix}x\\ y\end{bmatrix}\right)=\begin{bmatrix}2x\\ 3y\end{bmatrix}.$$
This transformation stretches the $x$-coordinate by a factor of $2$ and the $y$-coordinate by a factor of $3$. So the matrix gives a clear rule for how points move on the plane 📈.
How the Matrix Is Built
One powerful fact is that a linear transformation is completely determined by what it does to the standard basis vectors. In $\mathbb{R}^2$, the standard basis vectors are
$$\mathbf{e}_1=\begin{bmatrix}1\\0\end{bmatrix}, \qquad \mathbf{e}_2=\begin{bmatrix}0\\1\end{bmatrix}.$$
If you know $T(\mathbf{e}_1)$ and $T(\mathbf{e}_2)$, then you know the whole matrix. The matrix representation of $T$ in the standard basis is formed by putting these images as columns:
$$A=\begin{bmatrix}T(\mathbf{e}_1) & T(\mathbf{e}_2)\end{bmatrix}.$$
Why does this work? Any vector $\mathbf{x} \in \mathbb{R}^2$ can be written as
$$\mathbf{x}=x_1\mathbf{e}_1+x_2\mathbf{e}_2.$$
Because $T$ is linear,
$$T(\mathbf{x})=T(x_1\mathbf{e}_1+x_2\mathbf{e}_2)=x_1T(\mathbf{e}_1)+x_2T(\mathbf{e}_2).$$
That is exactly the same as multiplying the vector by the matrix whose columns are $T(\mathbf{e}_1)$ and $T(\mathbf{e}_2)$.
Example 1: A Stretching Transformation
Suppose
$$T\left(\begin{bmatrix}1\\0\end{bmatrix}\right)=\begin{bmatrix}4\\0\end{bmatrix}, \qquad T\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}0\\-2\end{bmatrix}.$$
Then the matrix is
$$A=\begin{bmatrix}4 & 0\\ 0 & -2\end{bmatrix}.$$
Now if
$$\mathbf{x}=\begin{bmatrix}3\\5\end{bmatrix},$$
then
$$T(\mathbf{x})=A\mathbf{x}=\begin{bmatrix}4 & 0\\ 0 & -2\end{bmatrix}\begin{bmatrix}3\\5\end{bmatrix}=\begin{bmatrix}12\\-10\end{bmatrix}.$$
So the vector moves to $\begin{bmatrix}12\\-10\end{bmatrix}$.
Common Types of Matrix Representations
Matrix representations can describe many kinds of geometric changes.
Scaling
A scaling changes size. For example,
$$\begin{bmatrix}k & 0\\0 & k\end{bmatrix}$$
multiplies every vector by $k$. If $k>1$, vectors get longer. If $0<k<1$, vectors shrink. If $k<0$, the transformation also reflects vectors through the origin.
Reflection
A reflection flips vectors across a line or plane. For instance,
$$\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}$$
reflects a vector across the $x$-axis because the $y$-coordinate changes sign.
Shear
A shear slants a shape without necessarily changing its area too much. A horizontal shear can be represented by
$$\begin{bmatrix}1 & k\\0 & 1\end{bmatrix}.$$
This sends
$$\begin{bmatrix}x\y\end{bmatrix} \mapsto \begin{bmatrix}x+ky\y\end{bmatrix}.$$
A square may become a slanted parallelogram after a shear ✏️.
Rotation
A rotation turns vectors around the origin. A counterclockwise rotation by an angle $\theta$ is represented by
$$\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}.$$
For example, a $90^\circ$ counterclockwise rotation uses
$$\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}.$$
This sends
$$\begin{bmatrix}1\\0\end{bmatrix} \mapsto \begin{bmatrix}0\\1\end{bmatrix}.$$
That means the vector on the positive $x$-axis moves to the positive $y$-axis.
Why Matrix Representations Matter
Matrix representations are useful because they let us compute efficiently and combine transformations easily. If one transformation is represented by $A$ and another by $B$, then applying $A$ first and then $B$ is represented by
$$BA.$$
The order matters because matrix multiplication is not commutative in general, which means
$$BA \neq AB$$
in many cases.
This is important in real life. In computer graphics, a shape may be rotated, then shifted, then scaled. Each step can be written as a matrix, and the whole process can be combined into one matrix. That makes calculations faster and easier for computers 🎮.
Matrix representations also help us study important properties of a transformation, such as:
- whether the transformation is one-to-one,
- whether it covers every vector in the output space,
- whether it preserves orientation,
- how it changes area or volume.
For example, the determinant of a $2\times 2$ matrix tells us the scaling factor for area. If
$$A=\begin{bmatrix}a & b\\ c & d\end{bmatrix},$$
then
$$\det(A)=ad-bc.$$
If $\det(A)=0$, the transformation squashes the plane into a lower-dimensional shape, such as a line. If $\det(A)\neq 0$, the transformation is invertible.
From Transformation to Matrix and Back
Sometimes you are given a transformation rule and must find its matrix. Other times you are given the matrix and must describe the transformation.
Finding the Matrix
Suppose a transformation $T$ in $\mathbb{R}^2$ satisfies
$$T\left(\begin{bmatrix}1\\0\end{bmatrix}\right)=\begin{bmatrix}2\\1\end{bmatrix}, \qquad T\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}-1\\3\end{bmatrix}.$$
Then the matrix is
$$A=\begin{bmatrix}2 & -1\\1 & 3\end{bmatrix}.$$
To check, apply it to a general vector
$$\begin{bmatrix}x\y\end{bmatrix}.$$
Then
$$A\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}2x-y\x+3y\end{bmatrix}.$$
So the transformation rule is
$$T\left(\begin{bmatrix}x\y\end{bmatrix}\right)=\begin{bmatrix}2x-y\x+3y\end{bmatrix}.$$
Describing the Transformation from the Matrix
If you see
$$A=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix},$$
you can tell that the matrix swaps coordinates:
$$\begin{bmatrix}x\y\end{bmatrix} \mapsto \begin{bmatrix}y\x\end{bmatrix}.$$
This is a reflection across the line $y=x$. The matrix tells the whole story.
Connection to the Bigger Picture
Matrix representations are a bridge between geometry and algebra. A linear transformation is a concept about moving vectors in a structured way. A matrix is a tool that stores that rule in numbers.
This connection matters because many later topics in linear algebra depend on it, such as:
- solving systems of linear equations,
- finding inverses of transformations,
- understanding eigenvalues and eigenvectors,
- changing coordinates and bases.
In fact, many systems of equations can be written as
$$A\mathbf{x}=\mathbf{b}.$$
So the same matrix ideas used for transformations also help solve practical problems like balancing chemical equations, modeling network flow, or planning computer animations.
Conclusion
Matrix representations are one of the most useful ideas in linear algebra because they turn linear transformations into something you can calculate with. students, if you understand how a matrix is built from basis vectors, how it acts on general vectors, and how it connects to geometric changes, then you have a strong foundation for the rest of the course. The big idea is simple but powerful: a matrix is the algebraic face of a linear transformation. Once you know the matrix, you know the transformation ✅.
Study Notes
- A linear transformation $T$ satisfies $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$.
- A matrix representation is a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.
- In the standard basis, the columns of $A$ are $T(\mathbf{e}_1), T(\mathbf{e}_2), \dots$.
- A matrix representation lets you compute the image of any vector using matrix multiplication.
- Common transformations include scaling, reflection, shear, and rotation.
- The order of combined transformations matters because matrix multiplication is usually not commutative.
- The determinant helps describe area scaling and whether a transformation is invertible.
- Matrix representations connect geometry, algebra, and applications in science and technology.