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Linear Algebra With Applications
Found in: Page 185
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

T denotes the space of infinity sequence of real numbers,T(f(t))=f(t2) from P to P.

The function T is linear but not isomorphism.

See the step by step solution

Step by Step Solution

Step 1: Determine the linearity of .

Consider the function T(f(t))=f(t2) from P to P.

A function D is called a linear transformation on R if the function D satisfies the following properties.

  1. D(x+y)=D(x)+D(y) for all x,yR.
  2. D(αx)=αD(x) for all constant αR.

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 P thenT(f(t))=f(t2) and T(g(t))=g(t2).

Substitute the value f(t2) for T(f(t)) and g(t2) for T (g(t)) in T(f(t))+T(g(t)) for and for in as follows.


Now, simplify role="math" localid="1659414024917" T(f+g(t))as follows.

T(f+g(t))=f+gt2 =ft2+gt2T(f+g(t))=Tft+Tgt

Assume fP and αR then .

Substitute the value αf(t2) for T(αf(t)) as follows.


As T(f+g(t))=T(f(t))+T(g(t)) and T(αf(t))=αT(f(t)) , by the definiti0on of linear transformation T is linear.

Step 2: Determine the isomorphism of .

Assume f(t)=t implies f(t2)=t2,, substitute the value t2 for f(t2) and t for f(t) in the equation T(f(t))=f(t2) as follows.


As the function T define from P to P, but tdoes not belong to P.

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

Hence, the transformation T(f(t))=f(t)+f''(t)+sin (t) is linear but not isomorphism.

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