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Chapter 6: Q6.3-25E (page 331)

Question 25: Let U be the \(8 \times 4\) matrix in Exercise 36 in Section 6.2. Find the closest point to \({\bf{y}} = \left( {1,1,1,1,1,1,1,1} \right)\) in Col U. Write the keystrokes or commands you use to solve this problem.

Short Answer

Expert verified

The closest point is \(\widehat {\bf{y}} = \left( {\begin{array}{*{20}{c}}{1.2}\\{.4}\\{1.2}\\{1.2}\\{.4}\\{1.2}\\{.4}\\{.4}\end{array}} \right)\).

Step by step solution

01

The Best Approximation Theorem

Consider \(W\) as a subspace of \({\mathbb{R}^n}\), and assume that y as any vector in \({\mathbb{R}^n}\) and \(\widehat {\bf{y}}\) as the orthogonal projection of y onto \(W\). Then, theclosest point in \(W\) to \({\bf{y}}\) is \(\widehat {\bf{y}}\), such that \(\left\| {{\bf{y}} - \widehat {\bf{y}}} \right\| < \left\| {{\bf{y}} - {\bf{v}}} \right\|\) for every \({\bf{v}}\) in \(W\) distinct from \(\widehat {\bf{y}}\).

02

The orthogonal projection \(\widehat {\bf{y}}\) of y

Theorem 6states that a \(m \times n\) matrix \(U\) contains orthonormal columnssuch that \({U^T}U = I\).

According to theorem 6, \(U\) contains orthonormal columns because \({U^T}U = {I_4}\).

The orthogonal projection \(\widehat {\bf{y}}\)of \({\bf{y}}\) onto Col U is known as the closest point to y in Col U.

03

Determine the closest point to y

Consider that \(U = \left( {\begin{array}{*{20}{c}}{ - 6}&{ - 3}&6&1\\{ - 1}&2&1&{ - 6}\\3&6&3&{ - 2}\\6&{ - 3}&6&{ - 1}\\2&{ - 1}&2&3\\{ - 3}&6&3&2\\{ - 2}&{ - 1}&2&{ - 3}\\1&2&1&6\end{array}} \right)\) as in Exercise 36.

Use the MATLAB code to compute \(\widehat {\bf{y}} = U{U^T}{\bf{y}}\) by theorem 10 as shown below:

\(\begin{array}{c} > > U = \left( \begin{array}{l} - 6\,\,\, - 3\,\,\,6\,\,\,1;\, - 1\,\,\,2\,\,\,\,1\,\,\, - 6;\,3\,\,\,6\,\,\,3\,\,\, - 2;\,\,6\,\,\, - 3\,\,\,6\,\,\, - 1;\,\\2\,\,\, - 1\,\,\,2\,\,\,\,3;\,\, - 3\,\,\,\,6\,\,\,3\,\,\,2;\,\, - 2\,\,\, - 1\,\,\,2\,\,\, - 3;1\,\,\,2\,\,\,1\,\,\,6\end{array} \right)\\ > > {\bf{y}} = \left( {1\,;\,1;\,\,\,1\,;\,\,1\,;\,\,1;\,\,\,1;\,\,\,1;\,\,\,1} \right)\\ > > \widehat {\bf{y}} = U * U' * {\bf{y}}\end{array}\)

\(\widehat {\bf{y}} = \left( {\begin{array}{*{20}{c}}{1.2}\\{.4}\\{1.2}\\{1.2}\\{.4}\\{1.2}\\{.4}\\{.4}\end{array}} \right)\)

Thus, the closest point is \(\widehat {\bf{y}} = \left( {\begin{array}{*{20}{c}}{1.2}\\{.4}\\{1.2}\\{1.2}\\{.4}\\{1.2}\\{.4}\\{.4}\end{array}} \right)\).

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Most popular questions from this chapter

In Exercises 1-4, find a least-sqaures solution of \(A{\bf{x}} = {\bf{b}}\) by (a) constructing a normal equations for \({\bf{\hat x}}\) and (b) solving for \({\bf{\hat x}}\).

4. \(A = \left( {\begin{aligned}{{}{}}{\bf{1}}&{\bf{3}}\\{\bf{1}}&{ - {\bf{1}}}\\{\bf{1}}&{\bf{1}}\end{aligned}} \right)\), \({\bf{b}} = \left( {\begin{aligned}{{}{}}{\bf{5}}\\{\bf{1}}\\{\bf{0}}\end{aligned}} \right)\)

Find an orthogonal basis for the column space of each matrix in Exercises 9-12.

9. \(\left[ {\begin{aligned}{{}{}}3&{ - 5}&1\\1&1&1\\{ - 1}&5&{ - 2}\\3&{ - 7}&8\end{aligned}} \right]\)

Question: In Exercises 1 and 2, you may assume that\(\left\{ {{{\bf{u}}_{\bf{1}}},...,{{\bf{u}}_{\bf{4}}}} \right\}\)is an orthogonal basis for\({\mathbb{R}^{\bf{4}}}\).

2.\({{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{\bf{1}}\\{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{ - {\bf{2}}}\\{\bf{1}}\\{ - {\bf{1}}}\\{\bf{1}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{3}}} = \left[ {\begin{aligned}{\bf{1}}\\{\bf{1}}\\{ - {\bf{2}}}\\{ - {\bf{1}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{4}}} = \left[ {\begin{aligned}{ - {\bf{1}}}\\{\bf{1}}\\{\bf{1}}\\{ - {\bf{2}}}\end{aligned}} \right]\),\({\bf{x}} = \left[ {\begin{aligned}{\bf{4}}\\{\bf{5}}\\{ - {\bf{3}}}\\{\bf{3}}\end{aligned}} \right]\)

Write v as the sum of two vectors, one in\({\bf{Span}}\left\{ {{{\bf{u}}_1}} \right\}\)and the other in\({\bf{Span}}\left\{ {{{\bf{u}}_2},{{\bf{u}}_3},{{\bf{u}}_{\bf{4}}}} \right\}\).

In Exercises 1-6, the given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

6. \(\left( {\begin{aligned}{{}}3\\{ - 1}\\2\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 5}\\9\\{ - 9}\\3\end{aligned}} \right)\)

In Exercises 1-4, find the equation \(y = {\beta _0} + {\beta _1}x\) of the least-square line that best fits the given data points.

  1. \(\left( {0,1} \right),\left( {1,1} \right),\left( {2,2} \right),\left( {3,2} \right)\)
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