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Suppose u, v E V. Prove that (u, v) = 0 if and only if ||u|| ≤ ||u + av|| for all a € F.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.5: Permutations And Inverses

Problem 4E: 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .

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From Linear Algebra Done Right by Axler, how do I prove this for both F=C and F=R? I need to know both proofs and the difference. Appreciate your help!

Suppose u, ve V. Prove that (u, v) = 0 if and only if
||u|| ≤ ||u + av||
for all a € F.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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