Q20E TRUE OR FALSE?聽The function聽T[... [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 2: Q20E (page 108)

TRUE OR FALSE?
The function $T [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} {(y + 1)}^{2} & {(y - 1)}^{2} \\ {(x - 3)}^{2} & {(x + 3)}^{2} \end{matrix}]$ is a linear transformation.

Short Answer

Expert verified

The given statement is false.

Step by step solution

Linear transformation

The system $T [\begin{matrix} x \\ y \end{matrix}]$ is said to be a linear transformation, if $T [\begin{matrix} x_{1} & + & x_{2} \\ y_{1} & + & y_{2} \end{matrix}] = T [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] + [\begin{matrix} x_{2} \\ y_{2} \end{matrix}]$ .

Compute the transformation of the system

Consider the function.

$T [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} {(y + 1)}^{2} & - {(y - 1)}^{2} \\ {(x - 3)}^{2} & - {(x + 3)}^{2} \end{matrix}]$

Consider the condition to check for the linear transformation.

Let, $x = x_{1} + x_{2}, = y = y_{1} + y_{2}$

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

Separate the variables.

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

Therefore, $T [\begin{matrix} x_{1} & + & x_{2} \\ y_{1} & + & y_{2} \end{matrix}] 鈮� T [\begin{matrix} x_{1} \\ y_{1} \end{matrix}] + T [\begin{matrix} x_{2} \\ y_{2} \end{matrix}]$ .

Hence, the given statement is false.

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91影视

TRUE OR FALSE?
The function $T [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} {(y + 1)}^{2} & {(y - 1)}^{2} \\ {(x - 3)}^{2} & {(x + 3)}^{2} \end{matrix}]$ is a linear transformation.

Short Answer

Step by step solution

Linear transformation

Compute the transformation of the system

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91影视

TRUE OR FALSE?The functionT[xy]=[y+12y-12x-32x+32] is a linear transformation.

Short Answer

Step by step solution

Linear transformation

Compute the transformation of the system

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Applied Mathematics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

TRUE OR FALSE?
The function $T [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} {(y + 1)}^{2} & {(y - 1)}^{2} \\ {(x - 3)}^{2} & {(x + 3)}^{2} \end{matrix}]$ is a linear transformation.