A [[Coordinate System]] for [[Vector|Vectors]] in a [[Linear Subspace|Subspace]] $A$ is constructed by any [[Basis]] $\mathcal B$ of a [[Linear Subspace|Subspace]]. Any [[Vector]] $\vec x\in A$ can be [[Linear Independence|uniquely]] written as a [[Linear Combination]] of $\mathcal B$. >[!example] Example: [[Standard Basis]] #### Notation Describing a point or vector can be explicitly annotated with a [[Basis Vectors|Basis]], >[!example] [[Vector]] written as a [[Linear Combination]] > $P = \mat{a\\b\\c}_{S} = a\hat i + b \hat j + c \hat k $ If there is not subscript present, it is implied that its [[Basis]] is the [[Standard Basis]]. $\huge \vec v_{\mathcal B} = \mat{ \mathcal{\vec B}_{1} & \mathcal{\vec B}_{2} & \dots& \mathcal{\vec B}_{n} }\vec v $ >[!tip] More Info: [[Change of Basis Matrix]] #### Incomplete Basis’ $\huge \begin{align} B &\in \Rn 3 \\ B &= \set{\vec u, \vec v, Q} \\ \\ \alpha: \mathbf{x} &= Q + t \vec u + s \vec v \\ \end{align}$ Even though $B$ is in $\Rn3$, it does not describe all of space in $\Rn3$ but only an *affine subspace* in $\Rn 3$.