Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (x + i y) = x^{2} + y^{2}$ from $ℂ$ to $ℂ$ .

The transformation $T (x + iy) = x^{2} + y^{2}$ is not a linear transformation.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Linear Transformation

Consider two linear spaces V and W. A transformation T is said to be a linear transformation if it satisfies the properties,

$T (f + g) = T (f) + T (g) T (k v) = k T (v)$

For all elements f,g of v and k is scalar.

An invertible linear transformation is called an isomorphism.

Step 2: Check whether the given transformation is a linear or not.

Consider the transformation $T (x + iy) = x^{2} + y^{2}$ , from $ℂ$ to $ℂ$ .

Check whether the transformation satisfies the below two conditions or not.

$1 . T (A + B) = T (A) + T (B) 2 . T (k A) = k T (A)$

Verify the first condition.

Let $A = x_{1} + i y_{1}$ and $B = x_{2} + i y_{2}$ be arbitrary complex numbers from $ℂ$ . Then,

$T (A + B) = T (x_{1} + i y_{1} + x_{2} + i y_{2}) = T ((x_{1} + x_{2}) + i (y_{1} + y_{2})) = {(x_{1} + x_{2})}^{2} + {(y_{1} + y_{2})}^{2} = x_{1}^{2} + x_{2}^{2} + 2 x_{1} x_{2} + y_{1}^{2} + y_{2}^{2} + 2 y_{1} y_{2} = (x_{1}^{2} + y_{1}^{2}) + (x_{2}^{2} + y_{2}^{2}) + 2 x_{1} x_{2} + 2 y_{1} y_{2} = T (A) + T (B) + 2 x_{1} x_{2} + 2 y_{1} y_{2}$

It is clear that, the first condition $T (A + B) \neq T (A) + T (B)$ is not satisfied.

Thus, T is not a linear transformation.

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Linear Algebra With Applications

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Short Answer

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (x + i y) = x^{2} + y^{2}$ from $ℂ$ to $ℂ$ .

Step by Step Solution

Step 1: Definition of Linear Transformation

Step 2: Check whether the given transformation is a linear or not.

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, T(x+iy)=x2+y2 from ℂto ℂ.

Step by Step Solution

Step 1: Definition of Linear Transformation

Step 2: Check whether the given transformation is a linear or not.

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Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (x + i y) = x^{2} + y^{2}$ from $ℂ$ to $ℂ$ .