A linear transformation is a map $L: V \to W$ between vector spaces $V$ and $W$ that respects the linear structure of $V$ and $W$. Specifically, if ${\bf x}$ and ${\bf y}$ are vectors in $V$ and if $c_1$ and $c_2$ are scalars, then

1. $L({\bf x} +{\bf y}) = L({\bf x}) + L({\bf y}). \qquad$ That is, $L$ sends sums to sums.
2. $L(c_1 {\bf x}) = c_1 L({\bf x}). \qquad$ That is, $L$ sends multiples to multiples.
3. $L(c_1 {\bf x} + c_2 {\bf y}) = c_1 L({\bf x}) + c_2 L({\bf y}). \qquad$ That is, $L$ sends linear combinations to linear combinations.

We really only need to check the first two properties, since the third follows directly from the first two. There are many, many examples of linear transformations. In the following video, we talk about five of them.

The simplest example of a linear transformation is matrix multiplication. If $V = {\bf R}^n$ and $W= {\bf R}^m$ and $A$ is an $m \times n$ matrix, set $L({\bf x})=A{\bf x}$. It turns out that every linear transformation from ${\bf R}^n$ to ${\bf R}^m$ can be written in this way, and that the formula for $A$ is $$ A = \begin{pmatrix} \phantom{a} &&& \cr L({\bf e}_1) & L({\bf e}_2) & \cdots & L({\bf e}_n) \cr \phantom{a}&&& \end{pmatrix}$$ In the following video we derive this formula and apply it to rotations in the plane.

Many geometric operations in the plane are described by $2 \times 2$ or $3 \times 3$ matrices. That's how simple (1990's style) computer graphics work. Modern graphics use similar ideas, only in 3 or 4 dimensions instead of 2.