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Linear Algebra With Applications
Found in: Page 184
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

Find the transformation is linear and determine whether the transformation is an isomorphism.

The solution T is not a liner transformation.

See the step by step solution

Step by Step Solution

Step1: Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

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

For all elements f,g of V and k is scalar.

A linear transformation T:VW is said to be an isomorphism if and only if ker(T)={0} and im(T)=W or localid="1659415485656" dim(V)=dim(W).

Step2: Explanation of the solution        

The given transformation as follows.

Tf=f't+t2 , from P2 to P2.

By using the definition of linear transformation as follows.


Now, to check the first condition as follows.

Consider the polynomials as follows.

f( x ) and g( x ) from P2 and k is scalar.

Now, simplify as follows.

Tf+g=f+g'+t2 =f'+g'+t2 f'+t2+g'+t2Tf+gTf+Tg

Thus, the function T is not a linear transformation.

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