Definition. linear transformation [br94]

    A mapping $A$ of a vector space $X$ into a vector space $Y$ is said to be a linear transformation if $$ A (\mathbf{x}_{1} + \mathbf{x}_{2}) = A \mathbf{x}_{1} + A \mathbf{x}_{2}, \quad A(c \mathbf{x}) = cA \mathbf{x} $$ for all $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x} \in X$ and all scalars $c$.

    Notation 1. some notations [br94A]

      We often writes $A \mathbf{x}$ insteand of $A(\mathbf{x})$ if $A$ is linear.