Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 300: 33

$T$ is a linear transformation from $R^2$ into $R^2$

Let $A$ be a $m \times n$ matrix. The function $T$, defined by $T(v)=Av$, is a linear transformation from $R^n$ into $R^m$. Here, since $A$ is a $2\times 2$ matrix, then $n=m=2$. So, $T$ is a linear transformation from $R^2$ into $R^2$.