Q18E Consider unit vectorsv鈬赌1,...,v... [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 8: Q18E (page 392)

Consider unit vectors ${\overset{鈬赌}{v}}_{1}, . . ., {\overset{鈬赌}{v}}_{n} in {鈩�}^{n}$ insuch that the angle between ${\overset{鈬赌}{v}}_{i}$ and ${\overset{鈬赌}{v}}_{j}$ is $60 掳 for all i 鈮� j . find the n - volume of the n - parallelepiped spanned by {\overset{鈬赌}{v}}_{1}, . . ., {\overset{鈬赌}{v}}_{n} . Hint : Let A = [{\overset{鈬赌}{v}}_{1}, . . ., {\overset{鈬赌}{v}}_{n}], and think about the {matrixA}^{T} A and its determinant . Exercise 17 is useful .$ for all.

Short Answer

Expert verified

The volume of the n-parallelepiped spanned by ${\overset{鈬赌}{v}}_{1}, . . ., {\overset{鈬赌}{v}}_{n} is V = {(\frac{1}{2})}^{\frac{n - 1}{2}} . {(\frac{n + 1}{2})}^{\frac{1}{2}}$ .

Step by step solution

Define symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that does not change when its transpose is calculated.
A symmetric matrix is defined as one whose transpose is identical to the matrix itself.
A square matrix of sizenxn is symmetric if $B^{T} = B$ .

Find the volume

The volume of n-parallelepiped spanned by is $A = {\overset{鈬赌}{v}}_{1}, . . ., {\overset{鈬赌}{v}}_{n}$

$V = {(\det [A^{t} . A])}^{1 / 2}$

Apply the given conditions. We obtain,

$A^{t} . A = [\begin{matrix} 1 & \frac{1}{2} & 鈥� & \frac{1}{2} \\ \frac{1}{2} & 1 & 鈥� & \frac{1}{2} \\ 鈰� & 鈰� & 鈰� \\ \frac{1}{2} & \frac{1}{2} & 鈥� & 1 \end{matrix}]$

The determinant of the given matrix is

$\det [A^{t} . A] = {(1 - \frac{1}{2})}^{n - 1} (\frac{n}{2} + 1 - \frac{1}{2}) = {(\frac{1}{2})}^{n - 1} . \frac{n + 1}{2} Hence the required volume is V = {(\frac{1}{2})}^{\frac{n + 1}{2}} . {(\frac{n + 1}{2})}^{\frac{1}{2}} .$

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91影视

Short Answer

Step by step solution

Define symmetric matrix

Find the volume

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91影视

Consider unit vectorsv鈬赌1,...,v鈬赌nin鈩�ninsuch that the angle between v鈬赌iand v鈬赌jis 60掳foralli鈮�j.findthen-volumeofthen-parallelepipedspannedbyv鈬赌1,...,v鈬赌n.Hint:LetA=[v鈬赌1,...,v鈬赌n],andthinkaboutthematrixATAanditsdeterminant.Exercise17isuseful.for all.

Short Answer

Step by step solution

Define symmetric matrix

Find the volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Mechanics Maths

Logic and Functions

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.