# Definition Any collection of $n$ [[Linear Independence|linearly independent]] [[Vector|vectors]] in the [[Tangent Space|tangent space]] $T_P$ is a [[basis]] for $T_P$. By choosing a [[basis]] in each $T_P \, \forall \, P \in M$, for some [[manifold]] $M$, we arrive at a [[basis]] for [[Vector Field|vector fields]]. > **Coordinate Basis** > If we have a coordinate system $\{x_i\}$ in a [[neighborhood]] $U$ of $P$, where $P$ is a point in a [[manifold]] $M$, then the [[Coordinate Transformations on Manifolds|coordinates]] define the *coordinate [[basis]]* $\{\partial/\partial x_i\}$ at all points in $U$. One could use another [[basis]], say $\{\pmb{e}_i\}$, where $i$ is a label that distinguishes the [[basis]] vectors from one another and does not denote a component. At a point $P$, an arbitrary vector $\pmb{V}$ can be written as: $ \pmb{V} = \sum V^i \frac{\partial}{\partial x^i} = \sum V'^i \pmb{e}_i $ Where $\{V^i\}$ are the components of $\pmb{V}$ in the [[basis]] $\{\partial/\partial x^i\}$ and $\{V'^i\}$ are the [[Components|components]] in the [[basis]] $\{\pmb{e}_i\}$. The components are related to one another by vector [[Basis Transformation]]. If $\pmb{V}$ and the bases $\{\partial/\partial x^i\}$ and $\{\pmb{e}_i\}$ are regarded as [[Vector Field|vector fields]] instead, then the components $\{V^i\}$ and $\{V'^i\}$ are of the field $V$ are *functions* on $M$. Note that above we assumed that the vectors $\{\partial/ \partial x^i\}$ of an arbitrary coordinate system are all [[Linear Independence|linearly independent]]. This is just a condition for the coordinates to be "good" coordinates, i.e. they provide a [[Types of Maps|bijection]] from some neighborhood $U$ of $P$ onto a region $V$ of $\mathbb{R}^n$.