Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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please answer the following question with the formula listed

**Question 11:** Let's try the Gram-Schmidt process of orthogonalization with the matrix: \[ \begin{bmatrix} 0.223 & 0.668 & 0.710 \\ -0.125 & 0.741 & 0.660 \\ 0.795 & -0.234 & 0.560 \\ \end{bmatrix} \] What is the y-coordinate of r3-prime? (Don't normalize r3-prime, simply follow the steps, do not use the cross product) Round your answer to 3 decimal places. [Answer box]
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Transcribed Image Text:**Question 11:** Let's try the Gram-Schmidt process of orthogonalization with the matrix: \[ \begin{bmatrix} 0.223 & 0.668 & 0.710 \\ -0.125 & 0.741 & 0.660 \\ 0.795 & -0.234 & 0.560 \\ \end{bmatrix} \] What is the y-coordinate of r3-prime? (Don't normalize r3-prime, simply follow the steps, do not use the cross product) Round your answer to 3 decimal places. [Answer box]
The text in the image appears to be a series of vector equations used in mathematics, possibly in the context of numerical methods or linear algebra. The equations are as follows: 1. \( \mathbf{r}_1' \gets \mathbf{r}_1 \), 2. \( \mathbf{r}_2' \gets \mathbf{r}_2 - \frac{\mathbf{r}_2 \cdot \mathbf{r}_1'}{\mathbf{r}_1' \cdot \mathbf{r}_1'} \mathbf{r}_1' \), 3. \( \mathbf{r}_3' \gets \mathbf{r}_3 - \frac{\mathbf{r}_3 \cdot \mathbf{r}_1'}{\mathbf{r}_1' \cdot \mathbf{r}_1'} \mathbf{r}_1' - \frac{\mathbf{r}_3 \cdot \mathbf{r}_2'}{\mathbf{r}_2' \cdot \mathbf{r}_2'} \mathbf{r}_2' \). Explanation: - These equations appear to describe a process of vector orthogonalization, possibly using the Gram-Schmidt process. - \( \mathbf{r}_1' \) is initialized as \( \mathbf{r}_1 \). - \( \mathbf{r}_2' \) is obtained by projecting \( \mathbf{r}_2 \) onto \( \mathbf{r}_1' \) and then subtracting this projection from \( \mathbf{r}_2 \), ensuring \( \mathbf{r}_2' \) is orthogonal to \( \mathbf{r}_1' \). - \( \mathbf{r}_3' \) is computed similarly by orthogonalizing \( \mathbf{r}_3 \) with respect to both \( \mathbf{r}_1' \) and \( \mathbf{r}_2' \), ensuring it is orthogonal to both. These steps are fundamental in creating an orthogonal set of vectors from a linearly independent set.
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Transcribed Image Text:The text in the image appears to be a series of vector equations used in mathematics, possibly in the context of numerical methods or linear algebra. The equations are as follows: 1. \( \mathbf{r}_1' \gets \mathbf{r}_1 \), 2. \( \mathbf{r}_2' \gets \mathbf{r}_2 - \frac{\mathbf{r}_2 \cdot \mathbf{r}_1'}{\mathbf{r}_1' \cdot \mathbf{r}_1'} \mathbf{r}_1' \), 3. \( \mathbf{r}_3' \gets \mathbf{r}_3 - \frac{\mathbf{r}_3 \cdot \mathbf{r}_1'}{\mathbf{r}_1' \cdot \mathbf{r}_1'} \mathbf{r}_1' - \frac{\mathbf{r}_3 \cdot \mathbf{r}_2'}{\mathbf{r}_2' \cdot \mathbf{r}_2'} \mathbf{r}_2' \). Explanation: - These equations appear to describe a process of vector orthogonalization, possibly using the Gram-Schmidt process. - \( \mathbf{r}_1' \) is initialized as \( \mathbf{r}_1 \). - \( \mathbf{r}_2' \) is obtained by projecting \( \mathbf{r}_2 \) onto \( \mathbf{r}_1' \) and then subtracting this projection from \( \mathbf{r}_2 \), ensuring \( \mathbf{r}_2' \) is orthogonal to \( \mathbf{r}_1' \). - \( \mathbf{r}_3' \) is computed similarly by orthogonalizing \( \mathbf{r}_3 \) with respect to both \( \mathbf{r}_1' \) and \( \mathbf{r}_2' \), ensuring it is orthogonal to both. These steps are fundamental in creating an orthogonal set of vectors from a linearly independent set.
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