Q3.1-22E For each matrix A in Exercise 17... [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 3: Q3.1-22E (page 119)

For each matrix A in Exercise 17 through 22, describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically (as a line, plane, etc. in ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$

Short Answer

Expert verified

Thus, the image of $T$ is spanned by the vector of $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ which geometrically represents a line in ${鈩�}^{3}$ .

Step by step solution

Obtain image of the transformation of the given matrix.

The given matrix is:

$A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$

The objective is to describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically.

Geometrical image of transformation.

The image of $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$ consists of all vectors of the form:

localid="1664184139110" $T [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = A [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = x_{1} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] + x_{2} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] + x_{3} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = (x_{1} + x_{2} + x_{3}) [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

Hence, the image of $T$ is spanned by the vector of $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ which is geometrically represents a line in ${鈩�}^{3}$ .

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91影视

For each matrix A in Exercise 17 through 22, describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically (as a line, plane, etc. in ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$

Short Answer

Step by step solution

Obtain image of the transformation of the given matrix.

Geometrical image of transformation.

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91影视

For each matrix A in Exercise 17 through 22, describe the image of the transformation T(x鈫�)=Ax鈫�geometrically (as a line, plane, etc. in 鈩�2and鈩�3.A=[111111111]

Short Answer

Step by step solution

Obtain image of the transformation of the given matrix.

Geometrical image of transformation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Statistics

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

For each matrix A in Exercise 17 through 22, describe the image of the transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ geometrically (as a line, plane, etc. in ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$