Gram-Schmidtin menetelmä $$$\left[\begin{array}{c}t\\0\end{array}\right]$$$, $$$\left[\begin{array}{c}- t\\t\end{array}\right]$$$ varten

Ortonormalisoi vektorijoukko $$$\mathbf{\vec{v_{1}}} = \left[\begin{array}{c}t\\0\end{array}\right]$$$, $$$\mathbf{\vec{v_{2}}} = \left[\begin{array}{c}- t\\t\end{array}\right]$$$ Gram-Schmidtin menetelmällä.

Gram-Schmidt-prosessin mukaan $$$\mathbf{\vec{u_{k}}} = \mathbf{\vec{v_{k}}} - \sum_{j=1}^{k - 1} \operatorname{proj}_{\mathbf{\vec{u_{j}}}}\left(\mathbf{\vec{v_{k}}}\right)$$$, jossa $$$\operatorname{proj}_{\mathbf{\vec{u_{j}}}}\left(\mathbf{\vec{v_{k}}}\right) = \frac{\mathbf{\vec{u_{j}}}\cdot \mathbf{\vec{v_{k}}}}{\mathbf{\left\lvert\vec{u_{j}}\right\rvert}^{2}} \mathbf{\vec{u_{j}}}$$$ on vektoriprojektio.

Normalisoitu vektori on $$$\mathbf{\vec{e_{k}}} = \frac{\mathbf{\vec{u_{k}}}}{\mathbf{\left\lvert\vec{u_{k}}\right\rvert}}$$$.

Vaihe 1

$$$\mathbf{\vec{u_{1}}} = \mathbf{\vec{v_{1}}} = \left[\begin{array}{c}t\\0\end{array}\right]$$$

$$$\mathbf{\vec{e_{1}}} = \frac{\mathbf{\vec{u_{1}}}}{\mathbf{\left\lvert\vec{u_{1}}\right\rvert}} = \left[\begin{array}{c}\frac{t}{\left|{t}\right|}\\0\end{array}\right]$$$ (vaiheiden osalta katso yksikkövektorilaskin.)

Vaihe 2

$$$\mathbf{\vec{u_{2}}} = \mathbf{\vec{v_{2}}} - \operatorname{proj}_{\mathbf{\vec{u_{1}}}}\left(\mathbf{\vec{v_{2}}}\right) = \left[\begin{array}{c}0\\t\end{array}\right]$$$ (vaiheista katso vektoriprojektion laskin ja vektorien vähennyslaskin).

$$$\mathbf{\vec{e_{2}}} = \frac{\mathbf{\vec{u_{2}}}}{\mathbf{\left\lvert\vec{u_{2}}\right\rvert}} = \left[\begin{array}{c}0\\\frac{t}{\left|{t}\right|}\end{array}\right]$$$ (vaiheiden osalta katso yksikkövektorilaskin.)

Ortonormaalien vektorien joukko on $$$\left\{\left[\begin{array}{c}\frac{t}{\left|{t}\right|}\\0\end{array}\right], \left[\begin{array}{c}0\\\frac{t}{\left|{t}\right|}\end{array}\right]\right\}$$$A.