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.
73. Is the linear transformation $T (f (t)) = \int_{0}^{t} f (x) d x$ an isomorphism from $P_{n - 1}$ to $Z_{n}$ ?

The linear transformation $T (f (t)) = \int_{0}^{t} f (x) dx$ is an isomorphism from $P_{n - 1}$ to $Z_{n}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Linear transformation.

Consider two linear spaces and . A function from to 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 V 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: Verification whether the linear transformation is isomorphism.

Consider the linear transformation $T : P_{n - 1} \to Z_{n}$ given by $T (f (t)) = \int_{0}^{t} f (x) d x$ .

Consider any non-zero polynomial,

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

Then,

n-1

$T (f (t)) = \int_{0}^{t} (a_{0} t + a_{1} t + . . . + a_{n - 1} t^{n - 1}) d t = {[a_{0} t + a_{1} \frac{t^{2}}{2} + . . . + a_{n - 1} \frac{t^{n}}{n}]}_{0}^{t}$

On applying the limits,

$T (f (t)) = a_{0} t + a_{1} \frac{t^{2}}{2} + . . . + a_{n - 1} \frac{t^{n}}{n}$

Since, it non-zero there exists localid="1659421702278" $0 \leq i \leq n - 1$ such that $a_{i} \neq 0$ .

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

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

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.
73. Is the linear transformation $T (f (t)) = \int_{0}^{t} f (x) d x$ an isomorphism from $P_{n - 1}$ to $Z_{n}$ ?

Step by Step Solution

Step 1: Definition of Linear transformation.

Step 2: Definition of Isomorphism.

Step 3: Verification whether the linear transformation is isomorphism.

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Linear Algebra With Applications

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Short Answer

In Exercise 72 through 74 , let Zn be the set of all polynomials of degree ≤n such that f(0) = 0. 73. Is the linear transformation T(ft)=∫0tf(x)dx an isomorphism from Pn-1 to Zn?

Step by Step Solution

Step 1: Definition of Linear transformation.

Step 2: Definition of Isomorphism.

Step 3: Verification whether the linear transformation is isomorphism.

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In Exercise 72 through 74 , let $Z_{n}$ be the set of all polynomials of degree $\leq n$ such that f(0) = 0.
73. Is the linear transformation $T (f (t)) = \int_{0}^{t} f (x) d x$ an isomorphism from $P_{n - 1}$ to $Z_{n}$ ?