Linear Transformation on Finite-Dimensional vector space over \mathbf{R}

I introduce linear transformation on finite-dimensional vector space over \mathbf{R}

Linear Transformation on finitie dimensional vector space

Let T: \mathbf{R}^n \to \mathbf{R}^m be a function between Euclidean spaces, where n, m \in \mathbb{N} . The function T is called a linear transformation if and only if there exists a matrix A \in \mathbf{R}^{m \times n } such that