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Chapter 6: Q6.2-27E (page 331)
Question: Let U be a square matrix with orthogonal columns. Explain why U is invertible. (Mention the theorem you use.)
It is proved that the matrix U is invertible.
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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.
\(\begin{array} w&\\ & {44}&{61}&{81}&{113}&{131} \\ \hline {\ln w}&\\vline & {3.78}&{4.11}&{4.39}&{4.73}&{4.88} \\ \hline p&\\vline & {91}&{98}&{103}&{110}&{112} \end{array}\)
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 \({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).)
Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be a linear transformation that preserves lengths; that is, \(\left\| {T\left( {\bf{x}} \right)} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
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. \(\left( {\begin{aligned}{{}{}}3\\{ - 4}\\5\end{aligned}} \right),\left( {\begin{aligned}{{}{}}{ - 3}\\{14}\\{ - 7}\end{aligned}} \right)\)
Compute the least-squares error associated with the least square solution found in Exercise 4.
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