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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

Define an isomorphism from P3 to R3.

The solution is not to define an isomorphism.

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 dim(V)=dim(W).

Step2: Explanation of the solution

Consider the transformation as follows.


Since,role="math" localid="1659418414775" dim(P3)=4 and dim(R3)=3

Therefore, dim(P3)dm(R3).

Any two finite dimensional spaces will be isomorphic if and only if they will have the same dimension.

Thus, there is not possible to define any isomorphism from P3 to R3.

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