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, $T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] f r o m P_{2} t o R^{2}$ .

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 T.

$[\begin{matrix} f & (7) \\ f & (11) \end{matrix}] f o r T (f (t)) a n d [\begin{matrix} g & (7) \\ g & (11) \end{matrix}] f o r T (g (t)) i n T (f (t)) + T (g (t)) a s f o l l o w s$ Consider the function $T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] f r o m P_{2} t o R^{2}$

A function is called a linear transformation on if the function 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 $f, g \in P_{2}$ then localid="1659410729239" $T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] a n d T (g (t)) = [\begin{matrix} g & (7) \\ g & (11) \end{matrix}]$ and .

Substitute the value $[\begin{matrix} f & (7) \\ f & (11) \end{matrix}] f o r T (f (t)) a n d [\begin{matrix} g & (7) \\ g & (11) \end{matrix}] f o r T (g (t)) i n T (f (t)) + T (g (t)) a s f o l l o w s$

$T (f (t)) + T (g (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] + [\begin{matrix} g & (7) \\ g & (11) \end{matrix}]$

Now, simplify $T (\{f + g\} (t))$ as follows.

localid="1659410186169" $T (\{f + g\} (t)) = [\begin{matrix} \{f + g\} & (7) \\ \{f + g\} & (11) \end{matrix}] = [\begin{matrix} f (7) + & g (7) \\ f (11) + & g (11) \end{matrix}] = [\begin{matrix} f (7) \\ f (11) \end{matrix}] + [\begin{matrix} g (7) \\ g (11) \end{matrix}] T (\{f + g\} (t)) = T (f (t)) + T (g (t))$ '

Assume localid="1659410277772" $f \in P_{2} a n d α \in R t h e n T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}]$ and then .

Substitute the value $[\begin{matrix} αf & (7) \\ αf & (11) \end{matrix}] f o r T (f (t))$ as follows.

$T (α f (t)) = [\begin{matrix} αf & (7) \\ αf & (11) \end{matrix}] = α [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] T (α f (t)) = α T (f (t))$

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

Step 2: Determine the isomorphism of T.

As the function T define from $P_{2} t o R^{2} a n d P_{2}$ is spanned by $\{1, t, t^{2}\}$ means dimension of $P_{2}$ is 3 and dimension of $R^{2}$ is 2.

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

Hence, the transformation $T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}]$ 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, $T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] f r o m P_{2} t o R^{2}$ .

Step by Step Solution

Step 1: Determine the linearity of T.

Step 2: Determine the isomorphism of T.

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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))=[f7f11] from P2 to R2 .

Step by Step Solution

Step 1: Determine the linearity of T.

Step 2: Determine the isomorphism of T.

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T denotes the space of infinity sequence of real numbers, $T (f (t)) = [\begin{matrix} f & (7) \\ f & (11) \end{matrix}] f r o m P_{2} t o R^{2}$ .