💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Stuck on your homework problem? This step-by-step video should help.

Try Numerade Free for 30 Days

Like

Report

Exercises $22-26$ provide a glimpse of some widely used matrix factorizations, some of which are discussed later in the text.(QR Factorization) Suppose $A=Q R,$ where $Q$ and $R$ are $n \times n, R$ is invertible and upper triangular, and $Q$ has the property that $Q^{T} Q=I .$ Show that for each $\mathbf{b}$ in $\mathbb{R}^{n},$ the equation $A \mathbf{x}=\mathbf{b}$ has a unique solution. What computations with $Q$ and $R$ will produce the solution?

$\mathbf{x}=\mathbf{R}^{-1} \mathbf{Q}^{\mathrm{T}} \mathbf{b}$

Algebra

Chapter 2

Matrix Algebra

Section 5

Matrix Factorizations

Introduction to Matrices

Missouri State University

Campbell University

Harvey Mudd College

Baylor University

Lectures

01:32

In mathematics, the absolu…

01:11

05:13

Exercises $22-26$ provide …

15:15

05:41

06:35

02:30

Exercises $27-29$ concern …

05:52

Chapter 7 will focus on ma…

06:33

Suppose $A$ is an $n \time…

01:51

Let $A$ be an $n \times n$…

02:10

In Exercises $13-20,$ find…

04:00

in this problem were introduced to the method of Q R factory ization. So I suppose that A is equal to Q times are where Q and R R n by n this means that, of course, is an n by N Matrix. Our is in vertebral and upper triangular, and Q has the property at its transposed because it's in verse so cute transpose times Q. Is the identity matrix, whereas to show that French column Vector B The equation X equals B has a unique solution. So because Q is square and transpose times, Q. Is equal toe. I. We have that Q is in vertebral by inverse matrix. The're, um, and we have the queue in verse. As I mentioned before, this is going to be cute transpose so it follows that A is the product of convertible matrices Q and R, and therefore it follows that a itself is in vertebral. So since a is in vertebral, it follows that the equation X equals B has a unique solution for X for all column vectors. Be now from our equation. X equals B. This implies that Q r X equals B, and so it follows that huge transpose Q r X equals Q transpose be so we have since Q. Transposed Q. Is the identity matrix I. R X equals Q transpose be. Or, in other words, our X equals q transposed be. And then we have that X is equal to are in verse. Que transpose be so That was part A or Sorry, not party. Now a good algorithm for finding X is to compute que transposed be then Reverend juice, the augmented matrix or huge transposed be. What will end up with is the identity matrix on the left and are inverse que transpose b which is X on the right. And we had that The reduction is fast since matrix are is upper triangular. This is why we're choosing this method of computation. All we have to do is apply row replacement operations to the rose above each entry on the diagonal

View More Answers From This Book

Find Another Textbook

In mathematics, the absolute value or modulus |x| of a real number x is its …

Exercises $22-26$ provide a glimpse of some widely used matrix factorization…

Exercises $27-29$ concern an $m \times n$ matrix $A$ and what are often call…

Chapter 7 will focus on matrices $A$ with the property that $A^{T}=A$ . Exer…

Suppose $A$ is an $n \times n$ matrix with the property that the equation $A…

Let $A$ be an $n \times n$ invertible matrix. Show that the unique solution …

In Exercises $13-20,$ find an invertible matrix $P$ and a matrix $C$ of the …

02:01

Determine by inspection whether the vectors are linearly independent. Justif…

03:09

Suppose $A$ is an $m \times n$ matrix and there exist $n \times m$ matrices …

03:55

Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be an invertible linear …

10:29

Find the inverses of the matrices in Exercises 1$$\left[\begin{array…

09:09

In Exercises $1-6,$ solve the equation $A \mathbf{x}=\mathbf{b}$ by using th…

02:53

Suppose a linear transformation $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{…

02:00

Suppose $(B-C) D=0,$ where $B$ and $C$ are $m \times n$ matrices and $D$ is …

03:13

Exercises 1–4 refer to an economy that is divided into three sectors—manufac…

03:00

Find another set of equilibrium prices for the economy in Example $1 .$ Supp…

04:31

Let $A=\left[\begin{array}{rr}{3} & {-6} \\ {-1} & {2}\end{array}\ri…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.