Gram-Schmidtsches Orthonormalisierungsverfahren Beispiel Essay

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Einleitung

Das Gram-Schmidtsche Orthogonalisierungsverfahren erzeugt zu jedem System linear unabhängiger Vektoren ein Orthogonalsystem, das denselben Untervektorraum erzeugt.

Verfahren

Es seien die linear unabhängigen Vektoren \( w_1, \dots, w_n \) gegeben. Mit dem Verfahren kann nun ein Orthogonalsystem von \( n \) paarweise orthogonalen Vektoren berechnet werden, das denselben Untervektorraum erzeugt. Für die einzelnen Vektoren \( v_1, \dots, v_n \) gilt:

$$ \begin{aligned} v_1 &= w_1 \\ v_2 &= w_2 - \frac{\langle v_1, w_2\rangle}{\langle v_1, v_1\rangle} \, v_1 \\ v_3 &= w_3 - \frac{\langle v_1, w_3\rangle}{\langle v_1, v_1\rangle} \, v_1 - \frac{\langle v_2, w_3\rangle}{\langle v_2, v_2\rangle} \, v_2 \\ &\vdots \\ v_n &= w_n - \frac{\langle v_1, w_n\rangle}{\langle v_1, v_1\rangle} \, v_1 - \frac{\langle v_2, w_n\rangle}{\langle v_2, v_2\rangle} \, v_2 - \dots - \frac{\langle v_{n-1}, w_n\rangle}{\langle v_{n-1}, v_{n-1}\rangle} \, v_{n-1} = w_n - \sum_{i=1}^{n-1} \frac{\langle v_i, w_n\rangle}{\langle v_i, v_i\rangle} \, v_i \end{aligned} $$

Beispiel

Die folgenden linear unabhängigen Vektoren \( w_1 \) und \( w_2 \) erzeugen einen Untervektorraum.

$$ w_1 = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix},\quad w_2 = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} $$

Es werden nun zwei orthogonale Vektoren v1 und v2 berechnet, die denselben Untervektorraum erzeugen:

$$ \begin{aligned} v_1 &= w_1 = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} \\ v_2 &= w_2 - \frac{\langle v_1, w_2\rangle}{\langle v_1, v_1\rangle} \cdot v_1 = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} - \frac{12}{14} \cdot \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} = \frac{1}{7} \begin{pmatrix} -4 \\ 8 \\ 2 \end{pmatrix} \end{aligned} $$

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