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Chapter 6: Q2E (page 331)

Let \({\mathbb{R}^{\bf{2}}}\) have the inner product of Example 1. Show that the Cauchy-Schwarz inequality holds for \({\bf{x}} = \left( {{\bf{3}}, - {\bf{2}}} \right)\) and \({\bf{y}} = \left( { - {\bf{2}},{\bf{1}}} \right)\). (Suggestion: Study \({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).)

Short Answer

Expert verified

The Cauchy-Schwartz inequality is true.

Step by step solution

01

Find the values \(\left\| {\bf{x}} \right\|\) and \(\left\| {\bf{y}} \right\|\)

Find the value of \(\left\| {\bf{x}} \right\|\).

\(\begin{aligned}\left\| {\bf{x}} \right\| &= \sqrt {\left\langle {{\bf{x}},{\bf{x}}} \right\rangle } \\ &= \sqrt {\left\langle {\left( {3, - 2} \right),\left( { - 2,1} \right)} \right\rangle } \\ &= \sqrt {4\left( 3 \right)\left( 3 \right) + 5\left( { - 2} \right)\left( { - 2} \right)} \\ &= \sqrt {36 + 20} \\ &= \sqrt {56} \end{aligned}\)

Thus, the value of \(\left\| {\bf{x}} \right\|\) is \(\sqrt {56} \).

Find the value of \(\left\| {\bf{y}} \right\|\).

\(\begin{aligned}\left\| {\bf{y}} \right\| &= \sqrt {\left\langle {{\bf{y}},{\bf{y}}} \right\rangle } \\ &= \sqrt {\left\langle {\left( { - 2,1} \right),\left( { - 2,1} \right)} \right\rangle } \\ &= \sqrt {4\left( { - 2} \right)\left( { - 2} \right) + 5\left( 1 \right)\left( 1 \right)} \\ &= \sqrt {21} \end{aligned}\)

Thus, the value of \(\left\| {\bf{y}} \right\|\) is \(\sqrt {21} \).

02

Find the inner product and \({\left\| {\bf{x}} \right\|^{\bf{2}}}{\left\| {\bf{y}} \right\|^{\bf{2}}}\)

The inner product \(\left\langle {{\bf{x}},{\bf{y}}} \right\rangle \)can be calculated as follows:

\(\begin{aligned}\left\langle {{\bf{x}},{\bf{y}}} \right\rangle &= \left\langle {\left( {3, - 2} \right),\left( { - 2,1} \right)} \right\rangle \\ &= 4\left( 3 \right)\left( { - 2} \right) + 5\left( { - 2} \right)\left( 1 \right)\\ &= - 34\end{aligned}\)

Find the value of \({\left\| {\bf{x}} \right\|^2}{\left\| {\bf{y}} \right\|^2}\).

\(\begin{aligned}{\left\| {\bf{x}} \right\|^2}{\left\| {\bf{y}} \right\|^2} &= 56 \times 21\\ &= 1176\end{aligned}\)

03

Check for Cauchy-Schwartz inequality

By the Cauchy-Schwartz inequality:

\(\begin{aligned}\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right| \le \left\| {\bf{x}} \right\|\left\| {\bf{y}} \right\|\\{\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^2} \le {\left\| {\bf{x}} \right\|^2}{\left\| {\bf{y}} \right\|^2}\\1156 < 1176\end{aligned}\)

Thus, the Cauchy-Schwartz inequality is true.

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Most popular questions from this chapter

A healthy child鈥檚 systolic blood pressure (in millimetres of mercury) and weight (in pounds) are approximately related by the equation

\({\beta _0} + {\beta _1}\ln w = p\)

Use the following experimental data to estimate the systolic blood pressure of healthy child weighing 100 pounds.

\(\begin{array} w&\\ & {44}&{61}&{81}&{113}&{131} \\ \hline {\ln w}&\\vline & {3.78}&{4.11}&{4.39}&{4.73}&{4.88} \\ \hline p&\\vline & {91}&{98}&{103}&{110}&{112} \end{array}\)

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.

4. \(\left( {\begin{aligned}{{}{}}3\\{ - 4}\\5\end{aligned}} \right),\left( {\begin{aligned}{{}{}}{ - 3}\\{14}\\{ - 7}\end{aligned}} \right)\)

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

Exercises 13 and 14, the columns of \(Q\) were obtained by applying the Gram Schmidt process to the columns of \(A\). Find anupper triangular matrix \(R\) such that \(A = QR\). Check your work.

14.\(A = \left( {\begin{aligned}{{}{r}}{ - 2}&3\\5&7\\2&{ - 2}\\4&6\end{aligned}} \right)\), \(Q = \left( {\begin{aligned}{{}{r}}{\frac{{ - 2}}{7}}&{\frac{5}{7}}\\{\frac{5}{7}}&{\frac{2}{7}}\\{\frac{2}{7}}&{\frac{{ - 4}}{7}}\\{\frac{4}{7}}&{\frac{2}{7}}\end{aligned}} \right)\)

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.

3. \(\left( {\begin{aligned}{{}{}}2\\{ - 5}\\1\end{aligned}} \right),\left( {\begin{aligned}{{}{}}4\\{ - 1}\\2\end{aligned}} \right)\)
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