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19. Find the distance from to the line given by span 20. You showed earlier that- is an orthogonal set. Normalize the vectors in this set to create an orthonormal set.

19. Find the distance from to the line given by span 20. You showed earlier that- is an orthogonal set. Normalize the vectors in this set to create an orthonormal set.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need help with these two questions 19-20
### Vector Spaces and Orthogonality 16. **Orthogonality of Set B** Show that the set \( B = \left\{ \begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix} \right\} \) is an orthogonal set. 17. **Orthogonal Projection** Find the orthogonal projection of the vector \( x = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) onto span \( B \), where \( B \) is the set from problem 16. 18. **Vector Decomposition** Write \( \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix} \) as a sum of two vectors, where one vector is in span \( \begin{pmatrix} -5 \\ 4 \\ 0 \end{pmatrix} \) and the other vector is orthogonal to \( \begin{pmatrix} 5 \\ 4 \\ 0 \end{pmatrix} \). 19. **Distance to a Line** Find the distance from \( \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \) to the line given by span \( \begin{pmatrix} 2 \\ 3 \end{pmatrix} \). 20. **Normalization of an Orthogonal Set** You showed earlier that \( \left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ \sqrt{2} \end{pmatrix}, \begin{pmatrix} -\sqrt{2} \\ 1 \end{pmatrix} \right\} \) is an orthogonal set. Normalize the vectors in this set to create an orthonormal set.
Transcribed Image Text:### Vector Spaces and Orthogonality 16. **Orthogonality of Set B** Show that the set \( B = \left\{ \begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix} \right\} \) is an orthogonal set. 17. **Orthogonal Projection** Find the orthogonal projection of the vector \( x = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) onto span \( B \), where \( B \) is the set from problem 16. 18. **Vector Decomposition** Write \( \begin{pmatrix} 3 \\ 5 \\ -1 \end{pmatrix} \) as a sum of two vectors, where one vector is in span \( \begin{pmatrix} -5 \\ 4 \\ 0 \end{pmatrix} \) and the other vector is orthogonal to \( \begin{pmatrix} 5 \\ 4 \\ 0 \end{pmatrix} \). 19. **Distance to a Line** Find the distance from \( \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \) to the line given by span \( \begin{pmatrix} 2 \\ 3 \end{pmatrix} \). 20. **Normalization of an Orthogonal Set** You showed earlier that \( \left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ \sqrt{2} \end{pmatrix}, \begin{pmatrix} -\sqrt{2} \\ 1 \end{pmatrix} \right\} \) is an orthogonal set. Normalize the vectors in this set to create an orthonormal set.
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