Answer
See explanation
Work Step by Step
Concepts
Definition of Linear Transformation
Algebraic Properties of $\mathbb{R}^{n}$
Plan
Compute $T(S(c \mathbf{u}+d \mathbf{v}))$ for $\mathbf{u}, \mathbf{v}$ and scalars $c$ and $d$
Solve
Take $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{p}$ and let $c$ and $d$ be scalars. Then:
$T(S(c \mathbf{u}+d \mathbf{v}))=T(c \cdot S(\mathbf{u})+d \cdot S(\mathbf{v}))$ because $S$ is linear
$=c \cdot T(S((u))+d \cdot T(S(\mathbf{v})) \text { since } T \text { is linear } r$
Conclusion
The mapping $\mathbf{x} \rightarrow T(S(\mathbf{x}))$ is linear.
