A process of projection a [[Vector]] onto a [[Linear Subspace]]. >[!note] >A example of this is [[Vector Projection]] or projecting a point in $\R^3$ to its nearest [[Point]] on a plane $\alpha$. Given $\set{\vec u_{1}, \vec u_{2}, \vec u_{3},\dots}$ who is a [[Set|set]] of [[Linear Independence|Linearly Independent]] [[Vector|Vectors]], the [[Gram Schmitt Process]] produces a new [[Set|set]] of [[Vector|Vectors]] $\set{\vec v_{1},\vec v_{2}, \dots}$ that constitutes an [[Orthogonal Basis]] with the same [[Span]] as the original. $\huge \begin{align} \vec v_{1} &= \vec u_{1} \\ \vec v_{2} &= \vec u_{2} - \frac{\braket{\vec v_{1} , \vec u_{2}} }{ \braket{\vec v_{1},\vec v_{1} } } \vec v_{1} \\ \vec v_{3} &= \vec u_{3} - \frac{\braket{\vec v_{1},\vec u_{3} }}{\braket{\vec v_{1},\vec v_{1} }} \vec v_{1} - \frac{\braket{\vec v_{2},\vec u_{3} }}{\braket{\vec v_{2},\vec v_{2} }} \vec v_{2} \\ \vdots & \\ \vec v_{k} &= \vec u_{k} - \sum_{j<k} \frac{\braket{\vec v_{j},\vec u_{k} }}{\braket{\vec v_{j}, \vec v_{j} }} \vec v_{j} \end{align} $