Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T : V → W is linear, prove that (x, y), = (T (x),T (y)), defines an inner product on V if and only if T is one-to-one. %3D

Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T : V → W is linear, prove that (x, y), = (T (x),T (y)), defines an inner product on V if and only if T is one-to-one. %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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Let V be a vector space over R and W be an inner product space over R with inner
product (, ),. If T: V → W is linear, prove that (r, y), = (T (x),T (y)), defines an inner
product on V if and only if T is one-to-one.

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