/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q3E Exercise 3-8 refer to \({{\bf{P}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q3E (page 331)

Exercise 3-8 refer to \({{\bf{P}}_{\bf{2}}}\) with the inner product given by evaluation at \( - {\bf{1}}\), 0, and 1. (See Example 2).

3. Compute \(\left\langle {p,q} \right\rangle \), where \(p\left( t \right) = {\bf{4}} + t\), \(q\left( t \right) = {\bf{5}} - {\bf{4}}{t^{\bf{2}}}\).

Short Answer

Expert verified

The inner product is 28.

Step by step solution

01

Find the values of polynomials

The values of polynomial \(p\left( t \right) = 4 + t\) are:

\(\begin{aligned}p\left( { - 1} \right) &= 4 - 1\\ &= 3\end{aligned}\)

\(\begin{aligned}p\left( 0 \right) &= 4 + 0\\ &= 4\end{aligned}\)

\(\begin{aligned}p\left( 1 \right) &= 4 + 1\\ &= 5\end{aligned}\)

The values of polynomial \(q\left( t \right) = 5 - 4{t^2}\) are:

\(\begin{aligned}q\left( { - 1} \right) &= 5 - 4{\left( { - 1} \right)^2}\\ &= 5 - 4\\ &= 1\end{aligned}\)

\(\begin{aligned}q\left( 0 \right) &= 5 - 4{\left( 0 \right)^2}\\ &= 5\end{aligned}\)

\(\begin{aligned}q\left( 1 \right) &= 5 - 4{\left( 1 \right)^2}\\ &= 5 - 4\\ &= 1\end{aligned}\)

02

Find the value of inner product

The inner product for \(\left\langle {p,q} \right\rangle \) is defined as:

\(\left\langle {p,q} \right\rangle = p\left( {{t_0}} \right)q\left( {{t_0}} \right) + p\left( {{t_1}} \right)q\left( {{t_1}} \right) + p\left( {{t_2}} \right)q\left( {{t_2}} \right)\)

Substitute \({t_0} = - 1\), \({t_1} = 0\) and \({t_2} = 1\).

\(\begin{aligned}\left\langle {p,q} \right\rangle &= p\left( { - 1} \right)q\left( { - 1} \right) + p\left( 0 \right)q\left( 0 \right) + p\left( 1 \right)q\left( 1 \right)\\ &= \left( 3 \right)\left( 1 \right) + \left( 4 \right)\left( 5 \right) + \left( 5 \right)\left( 1 \right)\\ &= 3 + 20 + 5\\ &= 28\end{aligned}\)

Thus, the inner product is 28.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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 \({\mathbb{R}^n}\).

  1. Show that T also preserves orthogonality; that is, \(T\left( {\bf{x}} \right) \cdot T\left( {\bf{y}} \right) = 0\) whenever \({\bf{x}} \cdot {\bf{y}} = 0\).
  2. Show that the standard matrix of T is an orthogonal matrix.

In Exercises 3鈥�6, verify that\[\left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\]is an orthogonal set, and then find the orthogonal projection of\[{\bf{y}}\]onto Span\[\left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\].

5.\[y = \left[ {\begin{aligned}{ - 1}\\2\\6\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}3\\{ - 1}\\2\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}1\\{ - 1}\\{ - 2}\end{aligned}} \right]\]

Suppose radioactive substance A and B have decay constants of \(.02\) and \(.07\), respectively. If a mixture of these two substances at a time \(t = 0\) contains \({M_A}\) grams of \(A\) and \({M_B}\) grams of \(B\), then a model for the total amount of mixture present at time \(t\) is

\(y = {M_A}{e^{ - .02t}} + {M_B}{e^{ - .07t}}\) (6)

Suppose the initial amounts \({M_A}\) and are unknown, but a scientist is able to measure the total amounts present at several times and records the following points \(\left( {{t_i},{y_i}} \right):\left( {10,21.34} \right),\left( {11,20.68} \right),\left( {12,20.05} \right),\left( {14,18.87} \right)\) and \(\left( {15,18.30} \right)\).

a.Describe a linear model that can be used to estimate \({M_A}\) and \({M_B}\).

b. Find the least-squares curved based on (6).

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

11. \(\left( {\begin{aligned}{*{20}{c}}{\frac{7}{4}}\\{\frac{1}{2}}\\1\end{aligned}} \right)\)

Show that if an \(n \times n\) matrix satisfies \(\left( {U{\bf{x}}} \right) \cdot \left( {U{\bf{y}}} \right) = {\bf{x}} \cdot {\bf{y}}\) for all x and y in \({\mathbb{R}^n}\), then \(U\) is an orthogonal matrix.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.