Exercise 21

Let $V$ be an $n$-dimensional space and suppose that $\vec{x}\in\mathbb{R}^{n}$. Fix a basis $B$ for $V$ and consider the map $h_{\vec{x}}\colon V\to\mathbb{R}$ given $\vec{v}\mapsto\vec{x}\mathbin{\smash{\bullet}}{\rm Rep}_{B}(\vec{v})$ by the dot product.

• Show that this map is linear.

• Show that for any linear map $g\colon V\to\mathbb{R}$ there is an $\vec{x}\in\mathbb{R}^{n}$ such that $g=h_{\vec{x}}$.

• In the prior item we fixed the basis and varied the $\vec{x}$ to get all possible linear maps. Can we get all possible linear maps by fixing an $\vec{x}$ and varying the basis?