Q43E Consider two vectors聽v鈫� and w... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q43E (page 291)

Consider two vectors $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ in $R^{n}$ . Form the matrix $A = [\begin{matrix} \overset{鈫�}{v} & \overset{鈫�}{w} \end{matrix}]$ . Express $d e t (A^{T} A)$ in terms of $| |\overset{鈫�}{v}| |, | |\overset{鈫�}{w}| |$ , and $\overset{鈫�}{v} . \overset{鈫�}{w}$ . What can you say about the sign of the result?

Short Answer

Expert verified

Therefore, the determinant of $\det (A^{T} A)$ is given by,

$\det (A^{T} A) = {||\overset{鈫�}{v}||}^{2} {||\overset{鈫�}{w}||}^{2} - (\overset{鈫�}{v} 路 \overset{鈫�}{w})^{2} 猢� 0$

Step by step solution

Step by Step Solution: Step 1: Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Given

Given matrix,

$A = [\begin{matrix} \overset{鈫�}{v} & \overset{鈫�}{w} \end{matrix}]$

To find det(ATA)

Let $\overset{鈫�}{v}, \overset{鈫�}{w} 鈭� {鈩�}^{n}$ . If

$A = [\begin{matrix} \overset{鈫�}{v} & \overset{鈫�}{w} \end{matrix}]$

$A^{T} = [\begin{matrix} \overset{鈫�}{v} \\ \overset{鈫�}{w} \end{matrix}]$ then,

$A^{T} A = [\begin{matrix} \overset{鈫�}{v} 路 \overset{鈫�}{v} & \overset{鈫�}{v} 路 \overset{鈫�}{w} \\ \overset{鈫�}{v} 路 \overset{鈫�}{w} & \overset{鈫�}{w} 路 \overset{鈫�}{w} \end{matrix}]$

So, $\det (A^{T} A) = {| |\overset{鈫�}{v}| |}^{2} {| |\overset{鈫�}{w}| |}^{2} - (\overset{鈫�}{v} 路 \overset{鈫�}{w})^{2} 猢� 0$ , due to the Cauchy-Schwarz-Bunyakowski inequality.

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Step by Step Solution: Step 1: Definition

Given

To find det(ATA)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Geometry

Decision Maths

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Consider two vectorsv鈫� and w鈫� in Rn . Form the matrix A=[v鈫�w鈫�] . Expressdet(ATA) in terms of |v鈫�|,|w鈫�|, andv鈫�.w鈫�. What can you say about the sign of the result?

Short Answer

Step by step solution

Step by Step Solution: Step 1: Definition

Given

To find det(ATA)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Geometry

Decision Maths

Statistics

Calculus

Study anywhere. Anytime. Across all devices.