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span{v1,..., vk}, where {v;}{_1 are orthonormal vectors, be a linear sub- (b) Let V := space of Rd. What is the projection Py of a vector a E Rd onto V? i=1

span{v1,..., vk}, where {v;}{_1 are orthonormal vectors, be a linear sub- (b) Let V := space of Rd. What is the projection Py of a vector a E Rd onto V? i=1

Algebra and Trigonometry (6th Edition)
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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please send handwritten solution for part b
1. (a) Define what is meant by a “scalar product" in Rd. (b) Let V := span{v1,. .., vk}, where {v;}_, are orthonormal vectors, be a linear sub- space of Rd. What is the projection Py of a vector x € Rd onto V? i=1 ( )T and vz := (}, )T. Suppose V; := span{v;}, i = 1,2, and æo := (c) Let vi := (1, 1). Show that Pv,Pv,x = ( vi, v2)( v1, x)v2 for all x € R?. (d) Show that (Pv,Pv, )"x := Pv,Pv, o...o Pv,Pv, r = {v1, v2)²n-1(v1, x)v2 for all æ € n times R². (e) Let xn+1 := Pv, Pv, Tn, n e No. Compute x1 and x2. (f) Does the sequence {xn},€N0 Converge? If so, to which point? (g) Provide a mathematical proof for your answer in (f). (h) For general unit vectors vị and v2, give a condition on vị and v2 such that the sequence {xn}nƐNo Converges. (i) Using the above results, write a Matlab code for obtaining the limit of the sequence {xn}n€No•
Transcribed Image Text:1. (a) Define what is meant by a “scalar product" in Rd. (b) Let V := span{v1,. .., vk}, where {v;}_, are orthonormal vectors, be a linear sub- space of Rd. What is the projection Py of a vector x € Rd onto V? i=1 ( )T and vz := (}, )T. Suppose V; := span{v;}, i = 1,2, and æo := (c) Let vi := (1, 1). Show that Pv,Pv,x = ( vi, v2)( v1, x)v2 for all x € R?. (d) Show that (Pv,Pv, )"x := Pv,Pv, o...o Pv,Pv, r = {v1, v2)²n-1(v1, x)v2 for all æ € n times R². (e) Let xn+1 := Pv, Pv, Tn, n e No. Compute x1 and x2. (f) Does the sequence {xn},€N0 Converge? If so, to which point? (g) Provide a mathematical proof for your answer in (f). (h) For general unit vectors vị and v2, give a condition on vị and v2 such that the sequence {xn}nƐNo Converges. (i) Using the above results, write a Matlab code for obtaining the limit of the sequence {xn}n€No•
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