$\huge A = \underbrace{Q}_{\small\text{Orthogonal}} \circ\underbrace{R}_{\small\text{Upper Triangular}} $ A form of [[Decomposition]] for some [[Matrix]] composing of an [[Orthogonal Matrix]] and an [[Upper Triangular Matrix]]. $\large \begin{align} A &= QR \\ Q^{-1}A &= Q^{-1}QR\\ Q^{\intercal}A &= R \end{align} $ $\begin{align} \let A &= \mat{ 1 & 2 & 1 \\ 1 & 0 & 3 \\ -1 & 1 & 1} \\ \vec v_{1} &= \vec A_{1} = \mat{1\\1\\-1} \\ \vec v_{2} &= \vec A_{2} - \frac{\vec v ^\intercal \vec A_{2}\vec v_{2} }{ \vec v_{1}^\intercal \vec v_{1} } = \mat{ \frac{5}{3}\\ -\frac{1}{3}\\ \frac{4}{3} } \\ \vec v_{3} &= \vec A_{3} - \frac{\vec v_{1} ^\intercal \vec A_{2}\vec v_{1} }{ \vec v_{1}^\intercal \vec v_{1} } - \frac{\vec v_{2} ^\intercal \vec A_{3}\vec v_{2} }{ \vec v_{2}^\intercal \vec v_{2} } = \mat{ -\frac{5}{7}\\ \frac{15}{7} \\ \frac{10}{7} } \\ Q &= \mat{ \hat{v}_{1} & \hat{v}_{2} & \hat{v}_{3}} = \mat{ } & \dots \end{align} $ >[!note]- [[../Digipen/Alexander Young|Alexander Young]] Tidbit >This is how a lot of [[../Digipen/Webwork|Webwork]] questions are made to be nice and [[Integer]] pilled.