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, $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f r o m P_{2} t o R^{3} .$ .

The function T is linear and isomorphism.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the linearity of T.

Consider the function $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f r o m P_{2} t o R^{3} .$

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 $f, g \in P_{2} t$ then $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}]$ and $T (g (t)) = [\begin{matrix} g (5) \\ g (7) \\ g (11) \end{matrix}]$ .

Substitute the value role="math" localid="1659411768440" $[\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f o r T (f (t)) a n d [\begin{matrix} g (5) \\ g (7) \\ g (11) \end{matrix}] f o r T (g (t)) i n T (f (t)) + T (g (t))$ as follows.

$T (f (t)) + T (g (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] + [\begin{matrix} g (5) \\ g (7) \\ g (11) \end{matrix}]$

Step 2: Simplify further

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

$T (\{f + g\} (t)) = [\begin{matrix} \{f + g\} & (5) \\ \{f + g\} & (7) \\ \{f + g\} & (11) \end{matrix}] = [\begin{matrix} f (5) & g (5) \\ f (7) & g (7) \\ f (11) & g (11) \end{matrix}] = [\begin{matrix} \begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix} \end{matrix}] + [\begin{matrix} \begin{matrix} g (5) \\ g (7) \\ g (11) \end{matrix} \end{matrix}] T (\{f + g\} (t)) = T (f (t)) + T (g (t))$

Assume $f \in P_{2} a n d α \in R$ then role="math" localid="1659412647658" $T (f (t)) = [\begin{matrix} \begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix} \end{matrix}]$ .

Substitute the value $[\begin{matrix} \begin{matrix} α f (5) \\ α f (7) \\ α f (11) \end{matrix} \end{matrix}] f o r T (α f (t))$ for as follows.

$T (α f (t)) = [\begin{matrix} \begin{matrix} α f (5) \\ α f (7) \\ α f (11) \end{matrix} \end{matrix}] = α [\begin{matrix} \begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix} \end{matrix}] T (α f (t) = α T (f (t)$

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 3: Determine the isomorphism of .

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

By the definition of polynomial, every polynomial form $P_{2}$ is described in a unique way means for every $[\begin{matrix} \begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix} \end{matrix}] \in R^{3}$ there exist a unique $f (t) \in P_{2}$ such that $T (f (t)) = [\begin{matrix} \begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix} \end{matrix}]$ .

By the definition of isomorphism, the function is isomorphism.

Hence, the transformation localid="1662122176677" $T (f (t)) = [\begin{matrix} \begin{matrix} f (5) \\ f (7) \end{matrix} \end{matrix}]$ is linear and 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, $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f r o m P_{2} t o R^{3} .$ .

Step by Step Solution

Step 1: Determine the linearity of T.

Step 2: Simplify further

Step 3: 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(5)f(7)f(11)] from P2 to R3. .

Step by Step Solution

Step 1: Determine the linearity of T.

Step 2: Simplify further

Step 3: Determine the isomorphism of .

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T denotes the space of infinity sequence of real numbers, $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f r o m P_{2} t o R^{3} .$ .