Let $V$ be [[finite-dimensional|fin dim]] [[vector space]] with [[linear basis]] $\mathcal{B} = \boldsymbol{b}_{1:N}$. Then each $\boldsymbol{v} \in V$ uniquely written as $v^{n} \boldsymbol{b}_{n}$ for some $v^{1:N} \in \mathbb{F}$. We call $(v^{1:N})^{\top}\in \mathbb{F}^{N}$ be the **coordinate vector** of $\boldsymbol{v}$ wrt $\mathcal B$. Note the mapping from vectors to their coordinates is an [[invertible linear map]] $V \twoheadrightarrow \mathbb{F}^{N}$ by definition of [[linear basis]] (each function has a unique representation). The components (extracting each coefficient) form the [[dual basis]]. when $\boldsymbol{b}_{1:N}$ is [[orthonormal]] this mapping is called the [[Fourier transform]]. This provides a [[chart]] that makes $\mathbb{F}^{N}$ a [[manifold]]. See [[Fourier transform]]