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Chapter 6: Orthogonality and Least Squares

Expert-verified
Linear Algebra and its Applications
Pages: 331 - 394
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Illustration

214 Questions for Chapter 6: Orthogonality and Least Squares

  1. In Exercises 7–10, let\[W\]be the subspace spanned by the\[{\bf{u}}\]’s, and write y as the sum of a vector in\[W\]and a vector orthogonal to\[W\].

    Found on Page 331

  2. In Exercises 11 and 12, find the closest point to\[{\bf{y}}\]in the subspace\[W\]spanned by\[{{\bf{v}}_1}\], and\[{{\bf{v}}_2}\].

    Found on Page 331

  3. In Exercises 11 and 12, find the closest point to \[{\bf{y}}\] in the subspace \[W\] spanned by \[{{\bf{v}}_1}\], and \[{{\bf{v}}_2}\].

    Found on Page 331

  4. In Exercises 13 and 14, find the best approximation to\[{\bf{z}}\]by vectors of the form\[{c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2}\].

    Found on Page 331

  5. In Exercises 13 and 14, find the best approximation to\[{\bf{z}}\]by vectors of the form\[{c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2}\].

    Found on Page 331

  6. Question: Let\[y = \left[ {\begin{aligned}5\\{ - 9}\\5\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}{ - 3}\\{ - 5}\\1\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 3}\\2\\1\end{aligned}} \right]\]. Find the distance from\[y\]to plane in\[{\mathbb{R}^3}\]spanned by\[{{\bf{u}}_1}\], and\[{{\bf{u}}_2}\].

    Found on Page 331

  7. Question: Let\[y\],\[{{\bf{v}}_1}\], and\[{{\bf{v}}_2}\]be as in Exercise 12. Find the distance from\[{\bf{y}}\]to the subspace of\[{\mathbb{R}^4}\]spanned by\[{{\bf{v}}_1}\], and\[{{\bf{v}}_2}\].

    Found on Page 331

  8. Let \({\rm{y}} = \left( {\begin{aligned}{{}}4\\8\\1\end{aligned}} \right)\), \({{\bf{u}}_1} = \left( {\begin{aligned}{{}}{{2 \mathord{\left/ {\vphantom {2 3}} \right.} 3}}\\{{1 \mathord{\left/

    Found on Page 331

  9. Question: Let\(y = \left( {\begin{aligned}{}7\\9\end{aligned}} \right)\),\({{\bf{u}}_1} = \left( {\begin{aligned}{}{{1 \mathord{\left/

    Found on Page 331

  10. In exercises 1-6, determine which sets of vectors are orthogonal.

    Found on Page 331

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