Dual Spaces

Given a vector space $V$, the dual space, $V^{*}$, is the space of linear functionals on $V$. Therefore the dual space can be defined as $\Omega : V \longrightarrow \mathbb{F}^{1}$ where $\Omega$ are funcationals from the vectors in $V$ to scalars $\mathbb{F}^{1}$. $V^{*}$ is itself a vector space if it fulfills all the axioms of a vector space, which can do done by adding an addition and scalar multiplication operation.

Each element in $V$ has a corresponding element in $V^{*}$ that can be thought of as the funcational that takes that element to an element in $\mathbb{F}$. Equally, any basis in $V$ has a corresponding basis in $V^{*}$.

Dual spaces are often used when working with vectors in non-orthogonal reference frames. Consider a basis $\{e_1, e_2, \ldots, e_{n}\}$ in $V$ and a vector $\bm{a}~\in~V$ with linear decomposition $c_1 e_1 + c_2 e_2 + \ldots c_n e_n$. A basis $\{e^1, e^2, \ldots, e^{n}\}$ can be found in $V^{*}$ such that