Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

In Exercise 72 through 74, let $Z_{n}$ be the set of all polynomials of degree $\leq n$ such that f(0) = 0.
74. Define an isomorphism from $Z_{n}$ to $P_{n - 1}$ (think calculus!).

The transformation $T (f (t)) = f' (t)$ is an isomorphism from $Z_{n}$ to $P_{n - 1}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Linear transformation

Consider two linear spaces V and W. A function T from V to W is called linear transformation if

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

for all elements f and g of V and for all scalars .

If is finite dimensional, then

$\dim (V) = rank (T) + nullity (T) = \dim (imT) + \dim (kerT)$

Step 2: Definition of Isomorphism

An invertible linear transformation T is called an isomorphism.

Step 3: Verify whether the linear transformation is an isomorphism

Consider the linear transformation $T : Z_{n} \to P_{n - 1}$ given by $T (f (t)) = f' (t)$ .

Since, $f (t), g (t) \in Z_{n}$

$T (f (t) + g (t)) = T ((f + g) t) = (f + g)' t = f' (t) + g' (t) = T (f (t)) + T (g (t))$

Also, consider a scalar c, then

$T (c f (t)) = T ((c f) t) = (c f)' t = c f' (t) = c T (f (t))$

Therefore, T is a linear transformation.

Consider any non-zero polynomial,

$f (t) = a_{0} + a_{1} t + . . . + a_{n - 1} t^{n - 1} \in P_{n - 1}$

Then,

$T (f (t)) = f' (t) = a_{1} + 2 a_{2} + . . . + n a_{n} t^{n - 1}$

Since, it non-zero there exists $0 \leq i \leq n - 1$ such that $a_{i} \neq 0$ .

Thus, $T (f (t))$ is also a non-zero polynomial in $P_{n - 1}$ .

Hence, $\ker (T) = {0}$ .

Therefore, $\dim (Z_{n}) = n = \dim (p_{n - 1})$ .

Thus, T is an isomorphism.

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Linear Algebra With Applications

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Short Answer

In Exercise 72 through 74, let $Z_{n}$ be the set of all polynomials of degree $\leq n$ such that f(0) = 0.
74. Define an isomorphism from $Z_{n}$ to $P_{n - 1}$ (think calculus!).

Step by Step Solution

Step 1: Definition of Linear transformation

Step 2: Definition of Isomorphism

Step 3: Verify whether the linear transformation is an isomorphism

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Linear Algebra With Applications

Answers without the blur.

Short Answer

In Exercise 72 through 74, let Zn be the set of all polynomials of degree ≤n such that f(0) = 0. 74. Define an isomorphism from Zn to Pn-1 (think calculus!).

Step by Step Solution

Step 1: Definition of Linear transformation

Step 2: Definition of Isomorphism

Step 3: Verify whether the linear transformation is an isomorphism

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Pure Maths

In Exercise 72 through 74, let $Z_{n}$ be the set of all polynomials of degree $\leq n$ such that f(0) = 0.
74. Define an isomorphism from $Z_{n}$ to $P_{n - 1}$ (think calculus!).