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 a linear transformation and is an isomorphism.

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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 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,

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

It is clear that, the first condition $T (A + B) = T (A) + T (B)$ is satisfied.

Verify the second condition.

Let k be an arbitrary scalar, and $A \in ℂ$ as follows.

$T (k A) = T (k (x_{1} + i y_{1})) = i k (x_{1} + i y_{1}) = k (i (x_{1} + i y_{1})) = k T (x_{1} + i y_{1}) T (k A) = k T (A)$

It is clear that, the second condition $T (k A) = k T (A)$ is also satisfied.

Thus,T is a linear transformation.

Step 3: Properties of isomorphism

A linear transformation $T : V \to W$ is said to be an isomorphism if and only if $k e r (T) = {0}$ and $I m (T) = W$ .

Now, check whether ker (T) $= {0}$ .

According to the definition of the kernel of a transformation,

localid="1659422383534" $k e r (T) = {A \in ℂ, T (A) = 0}$ .

Consider a complex number A as A = x + iy

Then,

$T (A) = 0 T (x + i y) = 0 i (x + i y) = 0 i x (x + i y) = 0 i x + i^{2} y = 0 + 0 i i x - y = 0 + 0 i - y + i x = 0 + 0 i$

Comparing both sides, it can be concluded that x = 0 , y = 0

Clearly, $k e r (T) = {0}$ .

Now, check whether Im (T) $= ℂ$ .

According to the definition of the image of a transformation,

localid="1659416713747" $I m (T) = \{T (A) : A \in\}$

Let $A = x + i y$ is in .

Then,

$T (A) = T (x + i y) = i (x + i y) = i x + i^{2} y = i x - y = - y + i x$

, say (B = - Y + ix).

T (A) = B

It is clear that, $T (A) = - y + i x$ .

This means for any localid="1659422430378" $B \in ℂ$ there exists a localid="1659422483414" $A \in ℂ$ such that $T (A) = B$ .

So, Im (T) = $ℂ$

Therefore, the transformation T is an isomorphism.

Thus, the transformation T is a linear transformation and T is an isomorphism.

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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.

Step 3: Properties of isomorphism

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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+iy)=x2+y2 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.

Step 3: Properties of isomorphism

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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 $ℂ$ .