4.1 What is a Linear Transformation?
Introduction
Imagine grabbing a map and stretching it, or rotating a sheet of paper so it faces a different direction. Every point on the surface moves simultaneously and according to a consistent rule. That intuition is exactly what a linear transformation captures.
A linear transformation is a function that reshapes a vector space without bending it. Straight lines stay straight, the origin stays fixed, and parallel lines remain parallel. From computer graphics and physics simulations to the layers of a neural network, linear transformations are one of the most fundamental tools in mathematics.
Definition
A linear transformation is a function between two vector spaces,
\[T : \mathbb{R}^n \to \mathbb{R}^m\]that satisfies the following two conditions.
Condition 1 – Additivity:
\[T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})\]Condition 2 – Homogeneity (scalar multiplication is preserved):
\[T(c\vec{u}) = c\,T(\vec{u})\]These two conditions can be combined into a single statement:
\[T(c_1\vec{u} + c_2\vec{v}) = c_1\,T(\vec{u}) + c_2\,T(\vec{v})\]This property is called linearity.
A Non-Example
The function $f(x) = x + 3$ is not a linear transformation. Since $f(0) = 3 \neq 0$, the origin does not map to the origin.
Every linear transformation must satisfy $T(\vec{0}) = \vec{0}$.
Matrix Representation
Every linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ can be represented completely by a matrix:
\[T(\vec{x}) = A\vec{x}\]where $A$ is an $m \times n$ matrix.
This is a remarkably powerful fact. All the information about the transformation is packed into a single matrix.
Images of the Basis Vectors
Recall the standard basis vectors of $\mathbb{R}^2$:
\[\hat{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \hat{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\]Apply the linear transformation $T$ to each basis vector:
\[T(\hat{e}_1) = \begin{pmatrix} a \\ c \end{pmatrix}, \quad T(\hat{e}_2) = \begin{pmatrix} b \\ d \end{pmatrix}\]Then the matrix of the transformation is simply:
\[A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\]In other words, the columns of the matrix are the images of the basis vectors.
For an arbitrary vector $\vec{x} = \begin{pmatrix} x_1 \ x_2 \end{pmatrix}$:
\[T(\vec{x}) = x_1\,T(\hat{e}_1) + x_2\,T(\hat{e}_2) = x_1 \begin{pmatrix} a \\ c \end{pmatrix} + x_2 \begin{pmatrix} b \\ d \end{pmatrix} = A\vec{x}\]
Example: Transforming the Unit Square
The unit square has vertices at
\[\begin{pmatrix}0\\0\end{pmatrix},\quad \begin{pmatrix}1\\0\end{pmatrix},\quad \begin{pmatrix}1\\1\end{pmatrix},\quad \begin{pmatrix}0\\1\end{pmatrix}\]Apply the matrix $A = \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix}$:
\[\begin{pmatrix}0\\0\end{pmatrix} \to \begin{pmatrix}0\\0\end{pmatrix},\quad \begin{pmatrix}1\\0\end{pmatrix} \to \begin{pmatrix}2\\0\end{pmatrix},\quad \begin{pmatrix}1\\1\end{pmatrix} \to \begin{pmatrix}3\\1\end{pmatrix},\quad \begin{pmatrix}0\\1\end{pmatrix} \to \begin{pmatrix}1\\1\end{pmatrix}\]The square has been transformed into a parallelogram. Straight lines remain straight, and parallel sides remain parallel – this is the geometric essence of linearity.
Key Takeaways
| Concept | Description / Formula |
|---|---|
| Linear transformation | $T : \mathbb{R}^n \to \mathbb{R}^m$, preserves addition and scalar multiplication |
| Additivity | $T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v})$ |
| Homogeneity | $T(c\vec{u}) = c\,T(\vec{u})$ |
| Origin condition | $T(\vec{0}) = \vec{0}$ (necessary condition) |
| Matrix representation | $T(\vec{x}) = A\vec{x}$ (every linear transformation has a matrix) |
| Columns = images of basis | Each column of $A$ is $T(\hat{e}_i)$ |
| Geometric meaning | Lines map to lines, parallel lines stay parallel, squares become parallelograms |
In the next post, we look at specific types of 2D linear transformations – rotation, reflection, shear, and scaling.