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 T is not a liner transformation.

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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.

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

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 $k e r (T) = {0}$ and $i m (T) = W$ or localid="1659415485656" $d i m (V) = d i m (W)$ .

Step2: Explanation of the solution

The given transformation as follows.

$T (f) = f' (t) + t^{2}$ , from $P_{2}$ to $P_{2}$ .

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 the polynomials as follows.

f( x ) and g( x ) from $P_{2}$ and k is scalar.

Now, simplify as follows.

$T (f + g) = (f + g)' + t^{2} = f' + g' + t^{2} \neq f' + t^{2} + g' + t^{2} T (f + g) \neq T (f) + T (g)$

Thus, the function T is not a linear transformation.

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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

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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

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