Q41E Find the matrix A聽of the orthog... [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 5: Q41E (page 234)

Find the matrix Aof the orthogonal projection onto the line in $R^{n}$ spanned by the vector
$\begin{array}{r} [\begin{matrix} 1 \\ 1 \\ . \\ . \\ . \\ 1 \end{matrix}] \end{array}}$ all ncomponents are 1.

Short Answer

Expert verified

The matrix of orthogonal projection is $\frac{1}{n} [\begin{matrix} 1 & 1 & 1 & . . . & 1 \\ 1 & 1 & 1 & . . . & 1 \\ . & . & . & . . . & . \\ . & . & . & . . . & . \\ 1 & 1 & 1 & . . . & 1 \end{matrix}]$ .

Step by step solution

Consider the theorem below.

The matrix P of the orthogonal projection onto V which has an orthonormal basis given by is given ${{\overset{鈯�}{u}}_{1}, {\overset{鈯�}{u}}_{2} . . . . . . {\overset{鈯�}{u}}_{m}} by P = Q Q^{T}, where Q = {{\overset{鈯�}{u}}_{1}, {\overset{鈯�}{u}}_{2} . . . . . . {\overset{鈯�}{u}}_{m}} .$

It wants to find out the matrix of the orthogonal projection onto the spanning set, say W.

Observe that the given vector is not of unit length, so we need to divide by its length. Thus,

$\overset{r}{u} = \frac{1}{\sqrt{n}} [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}]$ .

The required matrix.

By using the formula given in the above theorem the required matrix is,

$P = Q Q^{T} = \frac{1}{n} [\begin{matrix} 1 \\ 1 \\ . \\ . \\ 1 \end{matrix}] [\begin{matrix} 1 & 1 & 1 & . . . & 1 \end{matrix}] = \frac{1}{n} [\begin{matrix} 1 & 1 & 1 & . . . & 1 \\ 1 & 1 & 1 & . . . & 1 \\ . & . & . & . . . & . \\ . & . & . & . . . & . \\ 1 & 1 & 1 & . . . & 1 \end{matrix}] Hence, the required matrix of orthogonal projection will be \frac{1}{n} {[\begin{matrix} 1 & 1 & 1 & . . . & 1 \\ 1 & 1 & 1 & . . . & 1 \\ . & . & . & . . . & . \\ . & . & . & . . . & . \\ 1 & 1 & 1 & . . . & 1 \end{matrix}]}_{n 脳 n}$

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

Find the matrix Aof the orthogonal projection onto the line in $R^{n}$ spanned by the vector
$\begin{array}{r} [\begin{matrix} 1 \\ 1 \\ . \\ . \\ . \\ 1 \end{matrix}] \end{array}}$ all ncomponents are 1.

Short Answer

Step by step solution

Consider the theorem below.

The required matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Find the matrix Aof the orthogonal projection onto the line inRnspanned by the vector11...1}all ncomponents are 1.

Short Answer

Step by step solution

Consider the theorem below.

The required matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the matrix Aof the orthogonal projection onto the line in $R^{n}$ spanned by the vector
$\begin{array}{r} [\begin{matrix} 1 \\ 1 \\ . \\ . \\ . \\ 1 \end{matrix}] \end{array}}$ all ncomponents are 1.