Linear vs. non-linear are two different types of transformations. Here gives the details of the linear transformation.
A linear transformation between two vector spaces $V$ and $W$ is a map $T:V->W$ such that the following hold:
- $T(v_1+v_2)=T(v_1)+T(v_2)$ for any vectors $v_1$ and $v_2$ in $V$, and
- $T(\alpha v)= \alpha T(v)$ for any scalar alpha.
For example, in dimension reduction domain, principal component analysis (PCA) is a linear transformation. And kernel PCA is a non-linear one.
Here are details (thanks @whuber for the suggestion).
Suppose we have data matrix $X$ and the Singular-value decomposition of $X$ is $X=UDV^T$, the PCA transformation is just a matrix multiplication on $X$, which is $XV$.
We can show the matrix multiplication is linear algebraically: Define $T(X)=XV$ (the transformation of $X$ is multiply $X$ by matrix $V$), then we know $(X_1+X_2)V=X_1V+X_2V$ and $\alpha X V=\alpha XV$.
Intrusively, you can think about linear transformation is shifting and stretching the data, and non-linear transformation will make more dramatic changes on data such as making data "inside out".
If we visualize the transformation, linear transformation looks like this
Nonlinear transformation looks like this