Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the transformation is linear and determine whether the transformation is an isomorphism.

The solution is that T is a liner transformation and not an isomorphism.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

$T (f + g) = T (f) + T (g) T (kf) = kT (f)$

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

A linear transformation $T : V \to W$ is said to be an isomorphism if and only if $\ker (T) = {0}$ and $im (T) = W$ or $\dim (V) = \dim (W)$ .

Step2: Explanation of the solution

The given transformation as follows.

$T (x_{0,} x_{1}, x_{2}, x_{3}, . . .) = (x_{0,} x_{2}, x_{4}, . . .)$ , from V to V.

By using the definition of linear transformation as follows.

$T (A + B) = T (A) + T (B) T (kA) = kT (A)$

Now, to check the first condition as follows.

Consider two infinite sequences V from as follows.

$x = (x_{0} . x_{1}, . . .)$ and $y = (y_{0} . y_{1}, . . .)$ be an infinite sequences from V.

Now, simplify as follows.

$T (x + y) = T (x_{0} + y_{0}, x_{1} + y_{1} + . . .) = (x_{0} + y_{0}, x_{2} + y_{2} + . . .) = (x_{0} + x_{2}, . . .) + (y_{0} + y_{2}, . . .) T (x + y) = T (x) + T (y)$

Simplify for the second condition as follows.

$T (kx) = T ({kx}_{0}, {kx}_{1}, {kx}_{2} + . . .) = ({kx}_{0}, {kx}_{2}, {kx}_{4} + . . .) = k (x_{0}, x_{2}, x_{4}) T (kx) = kT (x)$

Thus, T is a linear transformation.

Step3: Properties of isomorphism

A linear transformation $T : V \to W$ is isomorphism if and only if $\ker (t) = {0}$ and $lm (t) = W$

Now, it easy to show that $T (0, 1, 0, 1, . . .) = (0, 0, 0, . . .)$ and $T ((0, 2, 0, 2, . . .)) = (0, 0, 0, . . .)$ .

That is the inverse of role="math" localid="1659416754005" $(0, 0, 0, \dots)$ does not have unique vector to map to.

This means that T is not an isomorphism.

Thus, T is a linear transformation and not an isomorphism.

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Linear Algebra With Applications

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Short Answer

Find the transformation is linear and determine whether the transformation is an isomorphism.

Step by Step Solution

Step1: Definition of Linear Transformation

Step2: Explanation of the solution

Step3: Properties of isomorphism

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Linear Algebra With Applications

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Short Answer

Find the transformation is linear and determine whether the transformation is an isomorphism.

Step by Step Solution

Step1: Definition of Linear Transformation

Step2: Explanation of the solution

Step3: Properties of isomorphism

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