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Linear Algebra With Applications
Found in: Page 186
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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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?

The linear transformation Tft=0tfxdx is an isomorphism from Pn-1 to Zn.

See the step by step solution

Step by Step Solution

Step 1: Definition of Linear transformation.

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

  1. T(f+g)=T(f)+T(g)
  2. T(kf)=kT(f)

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

If V is finite dimensional, then


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:Pn-1Zn given by Tft=0tfxdx.

Consider any non-zero polynomial,

ft=a0+a1t+...+an-1tn-1 Pn-1



Tft=0ta0t+a1t+...+an-1tn-1dt =a0t+a1t22+...+an-1tnn0t

On applying the limits,


Since, it non-zero there exists localid="1659421702278" 0in-1 such that ai0.

Thus, Tft is also a non-zero polynomial in Zn.

Also, ker(T)={0}.

Therefore, dimZn=n=dimPn-1.

Thus, T is an isomorphism.

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