Let \(T:{\mathbb{R}^p} \to {\mathbb{R}^n}\) and \(S:{\mathbb{R}^p} \to {\mathbb{R}^m}\) be linear transformations. Show that the mapping \({\bf{x}}| \to T\left( {S\left( {\bf{x}} \right)} \right)\) is a linear transformation (from \({\mathbb{R}^p}\) to \({\mathbb{R}^m}\)). (Hint:Compute \(T\left( {S\left( {c{\bf{u}} + d{\bf{v}}} \right)} \right)\) for u; v in \({\mathbb{R}^p}\) and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.)

Transformation \(T\) is said to be linear if all vectors \({\bf{u}}\) in \({\mathbb{R}^n}\) and all scalars \(c\) and \(d\) are represented in the domain \(T\), as shown below:

• \(T\left( {c{\bf{u}}} \right) = cT\left( {\bf{u}} \right)\)
• \(T\left( {c{\bf{u}} + d{\bf{v}}} \right) = cT\left( {\bf{u}} \right) + dT\left( {\bf{v}} \right)\)

The multiplication of matrix\(A\)of the order\(m \times n\)and vector x gives a new vector defined as\(A{\bf{x}}\)or b.

This concept is defined by the transformation rule \(T\left( {\bf{x}} \right)\). The matrix transformation is denoted as \({\bf{x}}| \to A{\bf{x}}\).

Let u and v be two vectors in\({\mathbb{R}^p}\), and c and d be the scalars.

It is given that \(T:{\mathbb{R}^p} \to {\mathbb{R}^n}\)and\(S:{\mathbb{R}^p} \to {\mathbb{R}^m}\)are in linear transformation.

Compute\(T\left( {S\left( {c{\bf{u}} + d{\bf{v}}} \right)} \right)\).

\(T\left( {S\left( {c{\bf{u}} + d{\bf{v}}} \right)} \right) = T\left( {c \cdot S\left( {\bf{u}} \right) + d \cdot S\left( {\bf{v}} \right)} \right)\)

Transform it further, as shown below.

\(\begin{aligned} T\left( {S\left( {c{\bf{u}} + d{\bf{v}}} \right)} \right) &= T\left( {c \cdot S\left( {\bf{u}} \right) + d \cdot S\left( {\bf{v}} \right)} \right)\\ &= c \cdot T\left( {S\left( {\bf{u}} \right)} \right) + d \cdot T\left( {S\left( {\bf{v}} \right)} \right)\end{aligned}\)

This shows that the above mapping is linear.

Thus, \({\bf{x}}| \to T\left( {S\left( {\bf{x}} \right)} \right)\) is a linear transformation.