In mathematics, a transformation is a function that takes an input and produces an output. You've seen functions like $f(x) = x^2$, which takes a number and outputs its square. A linear transformation does something similar, but its inputs and outputs are vectors.

Think of a transformation as a rule that moves, stretches, or rotates vectors. It takes an input vector and maps it to a new output vector in a specific, predictable way. Not just any transformation is linear, though. For a transformation to be linear, it must follow two very specific rules.

Let's unpack what

"preserving" these operations really means.

A transformation, let's call it $T$, is linear if it satisfies two conditions for any vectors $u$ and $v$, and any scalar $c$.

Imagine two vectors, $u$ and $v$. Their sum, $u+v$, forms the diagonal of a parallelogram. A linear transformation might stretch or rotate this parallelogram, but it will still be a parallelogram. The new diagonal will be the sum of the transformed sides, $T(u) + T(v)$.

If you double the length of a vector and then apply the transformation, the output vector will be double the length of the original transformed vector. The scaling factor can be passed through the transformation.

These two rules together ensure that grid lines in a vector space remain parallel and evenly spaced after the transformation. The origin $(0,0)$ also stays in place, since $T(0) = T(0 \cdot v) = 0 \cdot T(v) = 0$.

These rules might seem abstract, but they describe very concrete geometric operations. Common examples of linear transformations include rotations, reflections, and scaling.

For instance, rotating every vector in a 2D plane by 45 degrees counter-clockwise is a linear transformation. If you rotate two vectors and add them, you get the same result as if you add them first and then rotate their sum.

Another example is a reflection across an axis. Flipping every vector across the y-axis is a linear transformation because it preserves the parallelogram structure of vector addition and scaling relationships.

The most powerful way to work with linear transformations is by using matrices. Any linear transformation can be represented by a matrix. When you multiply a matrix by a vector, you are applying a linear transformation to that vector.

Let's say we have a transformation $T$ that takes a 2D vector $\begin{pmatrix} x \\ y \end{pmatrix}$ as input. The action of $T$ can be described by a 2x2 matrix $A$.

For example, the matrix that reflects vectors across the y-axis is:

If we apply this to a vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$:

The result is the vector $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$, which is the mirror image of $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ across the y-axis.

This connection is key. By encoding transformations as matrices, we can use the tools of matrix algebra to analyze and manipulate complex geometric operations. This sets the stage for understanding deeper properties of these transformations.