Exercise 27

Let $V$ be a vector space and assume that the maps $f_{1},f_{2}\colon V\to\mathbb{R}^{1}$ are linear.

• Define a map $F\colon V\to\mathbb{R}^{2}$ whose component functions are the given linear ones.

  $$\vec{v}\mapsto\begin{pmatrix}f_{1}(\vec{v})\\ f_{2}(\vec{v})\end{pmatrix}$$

  Show that $F$ is linear.

• Does the converse hold — is any linear map from $V$ to $\mathbb{R}^{2}$ made up of two linear component maps to $\mathbb{R}^{1}$?

• Generalize.