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Chapter 1: Q42E (page 1)

The two column vectors v鈬赌1and v鈬赌2 of a 2 x2 matrix A are shown in the accompanying figure. Let A=QR be the QR factorization of A. Represent the diagonal entries r11 and r22 of R as lengths in the figure. Interpret the product r11r22 as an area.

Short Answer

Expert verified

The product r11r22 is the area of the parallelogram.

Step by step solution

01

Determine QR factorization of matrix

In QR factorization of a matrix A, the matrix is decomposed into a product A = QR, where Q is an orthonormal matrix and R is an upper triangular matrix.

02

Represent the diagonal entries

Given that the length of the diagonal entries is equal to the column vector of the matrix A. So, r11=v鈬赌1andr22=v鈬赌2.

Now, the columns of the matrix Q can be computed as follows:

u1=1v鈬赌1v鈬赌1=1r11v鈬赌1

Also,

r22=vv鈬赌2鈯�

Here, r22 shows only the length of the normal vector v鈬赌2鈯�.

The representation of the diagonal entries as the area of the parallelogram is shown in the figure below:

The x-axis represents r11 and the y-axis represents r22.

Thus, the product r11r22 is the area of the parallelogram.

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Most popular questions from this chapter

Compute the products Ax in Exercises 16 through 19

using paper and pencil (if the products are defined).

16.

01322-3

Consider the function T(A)(X鈬赌)=x鈬赌TAx鈬赌fromRnxntoQn. Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.

Let A be a 4 脳 3 matrix, and letb鈫�and c鈫� be two vectors in 鈩�4. We are told that the systemrole="math" localid="1659341825668" Ax鈫�=b鈫� has a unique solution. What can you say about the number of solutions of the systemAx鈫�=c鈫� ?

In Exercises 1 through 12, find all solutions of the equations

with paper and pencil using Gauss鈥揓ordan elimination.

Show all your work.

|x1+2x3+4x4=-8x2-3x-x4=63x1+4x2-6x3+8x4=0-x2+3x3+4x4=-12|

in exercises 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

5.2x+3y=04x+5y=0
See all solutions

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