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Chapter 6: Q19E (page 331)

In Exercises 19 and 20, all vectors are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

  1. \(v \cdot {\rm{v}} = {\left\| {\rm{v}} \right\|^2}\).
  2. For any scalar \(c\),\({\rm{u}} \cdot \left( {c{\rm{v}}} \right) = c\left( {{\rm{u}} \cdot {\rm{v}}} \right)\).
  3. If the distance from \({\rm{u}}\) to \({\rm{v}}\) equals the distance from \({\rm{u}}\) to \( - {\rm{v}}\), then \({\rm{u}}\) and \({\rm{v}}\) are orthogonal.
  4. For a square matrix \(A\), vectors in \({\rm{Col }}A\) are orthogonal to vectors in Nul \(A\).
  5. If vectors \({{\rm{v}}_1}..........,{{\rm{v}}_p}\) span a subspace \(W\) and if \({\rm{x}}\) is orthogonal to each \({{\rm{v}}_j}\) for \(j = 1,..........,p\), then \({\rm{x}}\) is in \({W^ \bot }\)?.

Short Answer

Expert verified
  1. True, by using the definition of the length of the vector.
  2. True, by using Theorem 1(c).
  3. True, by using the definition of orthogonal vectors.
  4. False, as the case fails when the square matrix is not symmetric.
  5. True, by using the definition for the spans of any vector.

Step by step solution

01

Definition of Orthogonal Set

The two vectors \({\rm{u and v}}\) are Orthogonal if:

\(\begin{aligned}{l}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2}\\{\rm{and}}\\{\bf{u}} \cdot {\bf{v}} = 0\end{aligned}\).

02

 Verification of statement (a)

The definition for the length of any vector states that:

\({\left\| {\bf{v}} \right\|^2} = {\bf{v}} \cdot {\bf{v}}\)

Hence, the given statement is true.

03

 Verification of statement (b)

The Theorem 1(c)resembles that:

\(\left( {c{\bf{u}}} \right) \cdot {\bf{v}} = c\left( {{\bf{u}} \cdot {\bf{v}}} \right) = {\bf{u}} \cdot \left( {c{\bf{v}}} \right)\)

Hence, the given statement is true.

04

 Verification of statement (c)

The definition of the Orthogonal Vectorsstates that:

\(\begin{aligned}{l}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} = {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2}\\{\rm{and}}\\{\bf{u}} \cdot {\bf{v}} = 0\end{aligned}\)

Hence, the given statement is true.

05

 Verification of statement (d)

The given statement is only valid for the particular type of square matrices. Just in case if there is a matrix of type:

\(A = \left( {\begin{aligned}{*{20}{c}}1&1\\0&0\end{aligned}} \right)\)

The given statement would not be true.

Hence, the given statement is False.

06

 Verification of statement (e)

The definition for the spans of any vector states that for the given condition in question, the span:

\({W^ \bot } \subseteq {\mathbb{R}^n}\)

Hence, the given statement is true.

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Most popular questions from this chapter

A simple curve that often makes a good model for the variable costs of a company, a function of the sales level \(x\), has the form \(y = {\beta _1}x + {\beta _2}{x^2} + {\beta _3}{x^3}\). There is no constant term because fixed costs are not included.

a. Give the design matrix and the parameter vector for the linear model that leads to a least-squares fit of the equation above, with data \(\left( {{x_1},{y_1}} \right), \ldots ,\left( {{x_n},{y_n}} \right)\).

b. Find the least-squares curve of the form above to fit the data \(\left( {4,1.58} \right),\left( {6,2.08} \right),\left( {8,2.5} \right),\left( {10,2.8} \right),\left( {12,3.1} \right),\left( {14,3.4} \right),\left( {16,3.8} \right)\) and \(\left( {18,4.32} \right)\), with values in thousands. If possible, produce a graph that shows the data points and the graph of the cubic approximation.

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

13.\[z = \left[ {\begin{aligned}3\\{ - 7}\\2\\3\end{aligned}} \right]\],\[{{\bf{v}}_1} = \left[ {\begin{aligned}2\\{ - 1}\\{ - 3}\\1\end{aligned}} \right]\],\[{{\bf{v}}_2} = \left[ {\begin{aligned}1\\1\\0\\{ - 1}\end{aligned}} \right]\]

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

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

16. Use a matrix inverse to solve the system of equations in (7) and thereby obtain formulas for \({\hat \beta _0}\) , and that appear in many statistics texts.

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.

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

11. \(\left( {\begin{aligned}{{}{}}1&2&5\\{ - 1}&1&{ - 4}\\{ - 1}&4&{ - 3}\\1&{ - 4}&7\\1&2&1\end{aligned}} \right)\)
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