Given two vector spaces \(X\), \(Y\) over \(\mathbb{R}\) (or \(\mathbb{C}\)). A map \(T\colon X\to Y\) is called linear if for every \(x_1,x_2\in X\) and every \(\lambda\in \mathbb{R}\) (or every \(\lambda\in \mathbb{C}\)) we have
\[ T(\lambda x_1+x_2)=\lambda Tx_1+Tx_2. \]The map \(T\) is often called linear operator.