Q3.1-19E For each matrix in Exercise 17 t... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q3.1-19E (page 119)

For each matrix 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 ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 2 & 3 & 4 \\ - 2 & - 4 & - 6 & - 8 \end{matrix}]$

Short Answer

Expert verified

Thus, the image of is the line in of all scalar multiples of $[\begin{matrix} 1 \\ - 2 \end{matrix}]$ and is obtained.

Step by step solution

Obtain image of the transformation of the given matrix.

The given matrix is:

$A = [\begin{matrix} 1 & 2 & 3 & 4 \\ - 2 & - 4 & - 6 & - 8 \end{matrix}]$

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

The image of localid="1664178942436" $A = [\begin{matrix} 1 & 2 & 3 & 4 \\ - 2 & - 4 & - 6 & - 8 \end{matrix}]$ consists of all vectors of the form:

localid="1664179143946" $T [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = A [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix} 1 & 2 & 3 & 4 \\ - 2 & - 4 & - 6 & - 8 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = x_{1} [\begin{matrix} 1 \\ - 2 \end{matrix}] + x_{2} [\begin{matrix} 2 \\ - 4 \end{matrix}] + x_{3} [\begin{matrix} 3 \\ - 6 \end{matrix}] + x_{4} [\begin{matrix} 4 \\ - 8 \end{matrix}] = x_{1} [\begin{matrix} 1 \\ - 2 \end{matrix}] + 2 x_{2} [\begin{matrix} 1 \\ - 2 \end{matrix}] + 3 x_{2} [\begin{matrix} 1 \\ - 2 \end{matrix}] + 4 x_{4} [\begin{matrix} 1 \\ - 2 \end{matrix}]$

Solve further as:

$T [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = (x_{1} + 2 x_{2} + 3 x_{2} + 4 x_{4}) [\begin{matrix} 1 \\ - 2 \end{matrix}]$

Since, the vectors $[\begin{matrix} 1 \\ - 2 \end{matrix}], [\begin{matrix} 2 \\ - 4 \end{matrix}], [\begin{matrix} 3 \\ - 6 \end{matrix}]$ and $[\begin{matrix} 4 \\ - 8 \end{matrix}]$ are parallel, the image of $T$ is the line of all scalar multiples of $[\begin{matrix} 1 \\ - 2 \end{matrix}]$ .

Hence, the image of $T$ is the line in of all scalar multiples of $[\begin{matrix} 1 \\ - 2 \end{matrix}]$ .

Graph of image of transformation.

The following figure as shown below:

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91影视

For each matrix 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 ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 2 & 3 & 4 \\ - 2 & - 4 & - 6 & - 8 \end{matrix}]$

Short Answer

Step by step solution

Obtain image of the transformation of the given matrix.

Graph of image of transformation.

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91影视

For each matrix 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=[1234-2-4-6-8]

Short Answer

Step by step solution

Obtain image of the transformation of the given matrix.

Graph of image of transformation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Logic and Functions

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Study anywhere. Anytime. Across all devices.

For each matrix 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 ${鈩�}^{2} and {鈩�}^{3}$ .
$A = [\begin{matrix} 1 & 2 & 3 & 4 \\ - 2 & - 4 & - 6 & - 8 \end{matrix}]$