그람-슈미트 직교화 계산기

그람-슈미트 과정을 사용하여 벡터 $$$\mathbf{\vec{v_{1}}} = \left[\begin{array}{c}0\\3\\4\end{array}\right]$$$, $$$\mathbf{\vec{v_{2}}} = \left[\begin{array}{c}1\\0\\1\end{array}\right]$$$, $$$\mathbf{\vec{v_{3}}} = \left[\begin{array}{c}1\\1\\3\end{array}\right]$$$가 이루는 집합을 정규직교화하십시오.

그람-슈미트 과정에 따르면, $$$\mathbf{\vec{u_{k}}} = \mathbf{\vec{v_{k}}} - \sum_{j=1}^{k - 1} \operatorname{proj}_{\mathbf{\vec{u_{j}}}}\left(\mathbf{\vec{v_{k}}}\right)$$$이며, 여기서 $$$\operatorname{proj}_{\mathbf{\vec{u_{j}}}}\left(\mathbf{\vec{v_{k}}}\right) = \frac{\mathbf{\vec{u_{j}}}\cdot \mathbf{\vec{v_{k}}}}{\mathbf{\left\lvert\vec{u_{j}}\right\rvert}^{2}} \mathbf{\vec{u_{j}}}$$$는 벡터의 정사영이다.

정규화된 벡터는 $$$\mathbf{\vec{e_{k}}} = \frac{\mathbf{\vec{u_{k}}}}{\mathbf{\left\lvert\vec{u_{k}}\right\rvert}}$$$입니다.

1단계

$$$\mathbf{\vec{u_{1}}} = \mathbf{\vec{v_{1}}} = \left[\begin{array}{c}0\\3\\4\end{array}\right]$$$

$$$\mathbf{\vec{e_{1}}} = \frac{\mathbf{\vec{u_{1}}}}{\mathbf{\left\lvert\vec{u_{1}}\right\rvert}} = \left[\begin{array}{c}0\\\frac{3}{5}\\\frac{4}{5}\end{array}\right]$$$ (단계별 풀이는 단위 벡터 계산기를 참조하세요).

2단계

$$$\mathbf{\vec{u_{2}}} = \mathbf{\vec{v_{2}}} - \operatorname{proj}_{\mathbf{\vec{u_{1}}}}\left(\mathbf{\vec{v_{2}}}\right) = \left[\begin{array}{c}1\\- \frac{12}{25}\\\frac{9}{25}\end{array}\right]$$$ (단계는 벡터 사영 계산기 및 벡터 뺄셈 계산기를 참조하세요).

$$$\mathbf{\vec{e_{2}}} = \frac{\mathbf{\vec{u_{2}}}}{\mathbf{\left\lvert\vec{u_{2}}\right\rvert}} = \left[\begin{array}{c}\frac{5 \sqrt{34}}{34}\\- \frac{6 \sqrt{34}}{85}\\\frac{9 \sqrt{34}}{170}\end{array}\right]$$$ (단계별 풀이는 단위 벡터 계산기를 참조하세요).

3단계

$$$\mathbf{\vec{u_{3}}} = \mathbf{\vec{v_{3}}} - \operatorname{proj}_{\mathbf{\vec{u_{1}}}}\left(\mathbf{\vec{v_{3}}}\right) - \operatorname{proj}_{\mathbf{\vec{u_{2}}}}\left(\mathbf{\vec{v_{3}}}\right) = \left[\begin{array}{c}- \frac{3}{17}\\- \frac{4}{17}\\\frac{3}{17}\end{array}\right]$$$ (단계는 벡터 사영 계산기 및 벡터 뺄셈 계산기를 참조하세요).

$$$\mathbf{\vec{e_{3}}} = \frac{\mathbf{\vec{u_{3}}}}{\mathbf{\left\lvert\vec{u_{3}}\right\rvert}} = \left[\begin{array}{c}- \frac{3 \sqrt{34}}{34}\\- \frac{2 \sqrt{34}}{17}\\\frac{3 \sqrt{34}}{34}\end{array}\right]$$$ (단계별 풀이는 단위 벡터 계산기를 참조하세요).

직교정규 벡터들의 집합은 $$$\left\{\left[\begin{array}{c}0\\\frac{3}{5}\\\frac{4}{5}\end{array}\right], \left[\begin{array}{c}\frac{5 \sqrt{34}}{34}\\- \frac{6 \sqrt{34}}{85}\\\frac{9 \sqrt{34}}{170}\end{array}\right], \left[\begin{array}{c}- \frac{3 \sqrt{34}}{34}\\- \frac{2 \sqrt{34}}{17}\\\frac{3 \sqrt{34}}{34}\end{array}\right]\right\}\approx \left\{\left[\begin{array}{c}0\\0.6\\0.8\end{array}\right], \left[\begin{array}{c}0.857492925712544\\-0.411596604342021\\0.308697453256516\end{array}\right], \left[\begin{array}{c}-0.514495755427527\\-0.685994340570035\\0.514495755427527\end{array}\right]\right\}$$$A입니다.