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Chapter 6: Q8E (page 331)

Find an orthonormal basis of the subspace spanned by the vectors in Exercise 4.

Short Answer

Expert verified

An orthonormal basis is \(\left\{ {\left( {\begin{aligned}{{}{}}{\frac{3}{{\sqrt {50} }}}\\{\frac{{ - 4}}{{\sqrt {50} }}}\\{\frac{5}{{\sqrt {50} }}}\end{aligned}} \right),\left( {\begin{aligned}{{}{}}{\frac{1}{{\sqrt 6 }}}\\{\frac{2}{{\sqrt 6 }}}\\{\frac{1}{{\sqrt 6 }}}\end{aligned}} \right)} \right\}\).

Step by step solution

01

Compute \(\left\| {{{\bf{v}}_1}} \right\|\) and \(\left\| {{{\bf{v}}_2}} \right\|\)

Let the vectors \({{\bf{v}}_1} = \left( {\begin{aligned}{{}{}}3\\{ - 4}\\5\end{aligned}} \right),{{\bf{v}}_2} = \left( {\begin{aligned}{{}{}}3\\6\\3\end{aligned}} \right)\) from Exercise 4.

Compute \(\left\| {{{\bf{v}}_1}} \right\|\) and \(\left\| {{{\bf{v}}_2}} \right\|\) as shown below:

\(\begin{aligned}{}\left\| {{{\bf{v}}_1}} \right\| &= \sqrt {{3^2} + {{\left( { - 4} \right)}^2} + {5^2}} \\ &= \sqrt {9 + 16 + 25} \\ &= \sqrt {50} \\\left\| {{{\bf{v}}_2}} \right\| &= \sqrt {{3^2} + {6^2} + {3^2}} \\ &= \sqrt {9 + 36 + 9} \\ &= \sqrt {54} \\ &= 3\sqrt 6 \end{aligned}\)

02

Determine an orthonormal basis

Obtain an orthonormal basis as shown below:

\(\begin{aligned}{}\left\{ {\frac{{{{\bf{v}}_1}}}{{\left\| {{{\bf{v}}_1}} \right\|}},\frac{{{{\bf{v}}_2}}}{{\left\| {{{\bf{v}}_2}} \right\|}}} \right\} &= \left\{ {\frac{1}{{\sqrt {50} }}\left( {\begin{aligned}{{}{}}3\\{ - 4}\\5\end{aligned}} \right),\frac{1}{{3\sqrt 6 }}\left( {\begin{aligned}{{}{}}3\\6\\3\end{aligned}} \right)} \right\}\\ & = \left\{ {\left( {\begin{aligned}{{}{}}{\frac{3}{{\sqrt {50} }}}\\{\frac{{ - 4}}{{\sqrt {50} }}}\\{\frac{5}{{\sqrt {50} }}}\end{aligned}} \right),\left( {\begin{aligned}{{}{}}{\frac{1}{{\sqrt 6 }}}\\{\frac{2}{{\sqrt 6 }}}\\{\frac{1}{{\sqrt 6 }}}\end{aligned}} \right)} \right\}\end{aligned}\)

Hence, an orthonormal basis is \(\left\{ {\left( {\begin{aligned}{{}{}}{\frac{3}{{\sqrt {50} }}}\\{\frac{{ - 4}}{{\sqrt {50} }}}\\{\frac{5}{{\sqrt {50} }}}\end{aligned}} \right),\left( {\begin{aligned}{{}{}}{\frac{1}{{\sqrt 6 }}}\\{\frac{2}{{\sqrt 6 }}}\\{\frac{1}{{\sqrt 6 }}}\end{aligned}} \right)} \right\}\).

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Most popular questions from this chapter

Given data for a least-squares problem, \(\left( {{x_1},{y_1}} \right), \ldots ,\left( {{x_n},{y_n}} \right)\), the following abbreviations are helpful:

\(\begin{aligned}{l}\sum x = \sum\nolimits_{i = 1}^n {{x_i}} ,{\rm{ }}\sum {{x^2}} = \sum\nolimits_{i = 1}^n {x_i^2} ,\\\sum y = \sum\nolimits_{i = 1}^n {{y_i}} ,{\rm{ }}\sum {xy} = \sum\nolimits_{i = 1}^n {{x_i}{y_i}} \end{aligned}\)

The normal equations for a least-squares line \(y = {\hat \beta _0} + {\hat \beta _1}x\) may be written in the form

\(\begin{aligned}{c}{{\hat \beta }_0} + {{\hat \beta }_1}\sum x = \sum y \\{{\hat \beta }_0}\sum x + {{\hat \beta }_1}\sum {{x^2}} = \sum {xy} {\rm{ (7)}}\end{aligned}\)

Derive the normal equations (7) from the matrix form given in this section.

In Exercises 9-12, find a unit vector in the direction of the given vector.

10. \(\left( {\begin{aligned}{*{20}{c}}{ - 6}\\4\\{ - 3}\end{aligned}} \right)\)

In Exercises 9-12 find (a) the orthogonal projection of b onto \({\bf{Col}}A\) and (b) a least-squares solution of \(A{\bf{x}} = {\bf{b}}\).

12. \(A = \left[ {\begin{array}{{}{}}{\bf{1}}&{\bf{1}}&{\bf{0}}\\{\bf{1}}&{\bf{0}}&{ - {\bf{1}}}\\{\bf{0}}&{\bf{1}}&{\bf{1}}\\{ - {\bf{1}}}&{\bf{1}}&{ - {\bf{1}}}\end{array}} \right]\), \({\bf{b}} = \left( {\begin{array}{{}{}}{\bf{2}}\\{\bf{5}}\\{\bf{6}}\\{\bf{6}}\end{array}} \right)\)

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

11.\[y = \left[ {\begin{aligned}3\\1\\5\\1\end{aligned}} \right]\],\[{{\bf{v}}_1} = \left[ {\begin{aligned}3\\1\\{ - 1}\\1\end{aligned}} \right]\],\[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\{ - 1}\\1\\{ - 1}\end{aligned}} \right]\]

Let \({\mathbb{R}^{\bf{2}}}\) have the inner product of Example 1, and let \({\bf{x}} = \left( {{\bf{1}},{\bf{1}}} \right)\) and \({\bf{y}} = \left( {{\bf{5}}, - {\bf{1}}} \right)\).

a. Find\(\left\| {\bf{x}} \right\|\),\(\left\| {\bf{y}} \right\|\), and\({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).

b. Describe all vectors\(\left( {{z_{\bf{1}}},{z_{\bf{2}}}} \right)\), that are orthogonal to y.
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