Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(i) Let v, v' E V. Suppose that (vlr) = (vr) for every x V. Prove that v = v'. (ii) Suppose that S: V→W and T: W→ V are mappings that satisfies (Sv|w) = (v|Tw) (v EV, WEW) Note that the leff-hand side is the inner product in W (both Su and w are vectors in W), while the right-hand side is the inner product in V (both v and Tw are vectors in V). Prove that S is linear. 1

(i) Let v, v' E V. Suppose that (vlr) = (vr) for every x V. Prove that v = v'. (ii) Suppose that S: V→W and T: W→ V are mappings that satisfies (Sv|w) = (v|Tw) (v EV, WEW) Note that the leff-hand side is the inner product in W (both Su and w are vectors in W), while the right-hand side is the inner product in V (both v and Tw are vectors in V). Prove that S is linear. 1

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
See similar textbooks
icon
Related questions
Question
Let V, W be inner-product spaces.
(i) Let v, v' € V. Suppose that (vlx) = (vr) for every x V. Prove that v = v'. (ii) Suppose that S: VW and T: W→ V are mappings that satisfies (Sv|w) = (v|Tw) (v EV, WEW) Note that the leff-hand side is the inner product in W (both Su and w are vectors in W), while the right-hand side is the inner product in V (both v and Tw are vectors in V). Prove that S is linear. ¹
Transcribed Image Text:(i) Let v, v' € V. Suppose that (vlx) = (vr) for every x V. Prove that v = v'. (ii) Suppose that S: VW and T: W→ V are mappings that satisfies (Sv|w) = (v|Tw) (v EV, WEW) Note that the leff-hand side is the inner product in W (both Su and w are vectors in W), while the right-hand side is the inner product in V (both v and Tw are vectors in V). Prove that S is linear. ¹
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer