**Definition 0.0.1 **  A **linear transformation** is a function $T:V\to W$ for vector fields $V,W$ such that
1. $T(\mathbf{v}+\mathbf{w}) = T(\mathbf{v})+ T(\mathbf{w})$ for all $\mathbf{v},\mathbf{w}\in V$
2. $T(\alpha\mathbf{v}) = \alpha T(\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha$