Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

T denotes the space of infinity sequence of real numbers, $T (f (t)) = f (t^{2})$ from P to P.

The function T is linear but not isomorphism.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the linearity of .

Consider the function $T (f (t)) = f (t^{2})$ from P to P.

A function D is called a linear transformation on $R$ if the function D satisfies the following properties.

$D (x + y) = D (x) + D (y) f o r a l l x, y \in R .$
$D (α x) = α D (x) f o r a l l c o n s t a n t α \in R .$

An invertible linear transformation is called isomorphism or dimension of domain and co-domain is not same then the function is not isomorphism.

Assume $f, g \in P t h e n T (f (t)) = f (t^{2}) a n d T (g (t)) = g (t^{2}) .$

Substitute the value $f (t^{2}) f o r T (f (t)) a n d g (t^{2}) f o r T (g (t)) i n T (f (t)) + T (g (t))$ for and for in as follows.

$T (f (t)) + T (g (t)) = f (t^{2}) + g (t^{2})$

Now, simplify role="math" localid="1659414024917" $T (\{f + g\} (t))$ as follows.

$T (\{f + g\} (t)) = \{f + g\} {(t)}^{2} = f (t^{2}) + g (t^{2}) T (\{f + g\} (t)) = T (f (t)) + T (g (t))$

Assume $f \in P a n d α \in R t h e n .$

Substitute the value $α f (t^{2}) f o r T (α f (t))$ as follows.

$T (α f (t)) = α f (t^{2}) T (α f (t)) = α T (f (t))$

As $T (f + g (t)) = T (f (t)) + T (g (t)) a n d T (α f (t)) = α T (f (t))$ , by the definiti0on of linear transformation T is linear.

Step 2: Determine the isomorphism of .

Assume $f (t) = t i m p l i e s f (t^{2}) = t^{2},$ , substitute the value $t^{2} f o r f (t^{2}) a n d \sqrt{t} f o r f (t)$ in the equation $T (f (t)) = f (t^{2})$ as follows.

$T (f (t)) = f (t^{2}) T (\sqrt{t}) = t^{2}$

As the function T define from P to P, but $\sqrt{t}$ does not belong to P.

By the definition of isomorphism, the function T is not isomorphism.

Hence, the transformation $T (f (t)) = f (t) + f^{''} (t) + s i n (t)$ is linear but not isomorphism.

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Linear Algebra With Applications

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Short Answer

T denotes the space of infinity sequence of real numbers, $T (f (t)) = f (t^{2})$ from P to P.

Step by Step Solution

Step 1: Determine the linearity of .

Step 2: Determine the isomorphism of .

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Linear Algebra With Applications

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Short Answer

T denotes the space of infinity sequence of real numbers,T(f(t))=f(t2) from P to P.

Step by Step Solution

Step 1: Determine the linearity of .

Step 2: Determine the isomorphism of .

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T denotes the space of infinity sequence of real numbers, $T (f (t)) = f (t^{2})$ from P to P.