A linear transformation is a map between vector spaces that preserves their linear structure. A map

$$T:V\to W$$

is linear when

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

and

$$T(c\mathbf v)=cT(\mathbf v)$$

for all vectors and scalars.

Equivalently, linear transformations preserve every linear combination:

$$T(a\mathbf u+b\mathbf v)=aT(\mathbf u)+bT(\mathbf v).$$

The zero vector is fixed

Linearity forces

$$T(\mathbf0)=\mathbf0.$$

For this reason, a translation such as $T(x,y)=(x+1,y)$ is not linear: it moves the origin.

Geometric examples

Rotations about the origin, reflections through lines or planes through the origin, scalings and projections onto subspaces are linear transformations.

For example,

$$T(x,y)=(2x,2y)$$

scales every vector by a factor of two while preserving addition and scalar multiplication.

Knowing a basis is enough

If $\mathbf b_1,\ldots,\mathbf b_n$ form a basis and

$$\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n,$$

then linearity gives

$$T(\mathbf v)=c_1T(\mathbf b_1)+\cdots+c_nT(\mathbf b_n).$$

So a linear transformation is completely determined by what it does to a basis. This is what makes matrix representations possible.

Visual intuition: linear transformations