test Linear transformations and change of basis are widely used in statistics, for this reason I briefly describe the definition of these concepts and how they are related. Linear Transformation Letting \(V\) and \(W\) be vector spaces, a function \(f: V \rightarrow W\) is a linear transformation if the additivity and scalar multiplication properties are hold for any two vectors \(\mathbf{u}, \mathbf{v} \in V\) and a constant \(c\): \[f(\mathbf{u}+\mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})\] \[f(c\mathbf{v}) = cf(\mathbf{v}).