Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a linear map on V. In other words, show that if U is a subspace of V and S e L(U, W), then there exists T E L(V, W) such that Tu = Su for all u E U.

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Suppose V is finite-dimensional. Prove that every linear map on a
subspace of V can be extended to a linear map on V. In other words,
show that if U is a subspace of V and S e L(U, W), then there exists
TE L(V, W) such that Tu = Su for all u E U.
Su for all u e U.
Transcribed Image Text:Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a linear map on V. In other words, show that if U is a subspace of V and S e L(U, W), then there exists TE L(V, W) such that Tu = Su for all u E U. Su for all u e U.
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