Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 301: 59

False

A transformation is said to be linear if $T(av+w)=aT(v)+T(w)$, for all $ v,w$ in the given domain and for all scalars $a$. Observe that $D_x(sin(2x))=\displaystyle \frac{d(sin(2x))}{dx}=2cos(2x)$ and $2D_x(sin(x))=2\displaystyle \frac{d(sin(x)))}{dx}=2cos(x)$ We see that $D_x(sin(2x))=2{ }cos(2x)\neq 2cos(x=2D_x(sin(x)))$ Hence, the statement is false.