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Linear Algebra With Applications
Found in: Page 200
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


38. There exists a subspace of 3×4that is isomorphic toP9.

The given statement is true.

See the step by step solution

Step by Step Solution

Step 1: Isomorphism of vector space

Two vector spaces V and W over the same field F are isomorphic if there is a bisection T:VW which preserves addition and scalar multiplication, that is, for all vectors u and v in V, and all scalars cF,T(u+v)=T(u)+T(v) and T(cv)=cT(v) .The correspondence T is called an isomorphism of vector spaces.

When T:VW is an isomorphism,T:V'W then it’s emphasize that it is an isomorphism When V and W, are isomorphic, but the specific isomorphism is not named, we’ll just write V~=W..

Step 2: condition of a subspace to be isomorphic.

Space3×4 is 12-dimensional and P9 is 10-dimensional. Spaces of same dimension are isomorphic, so any 10-dimensional subspace of 3×4is isomorphic to P9.

Hence, the statement is true.

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