(a) Define what is meant by a "scalar product" in Rº. span{v1,..., vk}, where {v;}_, are orthonormal vectors, be a linear sub- (b) Let V := space of Rd. What is the projection Py of a vector x E Rd onto V? i=1 (c) Let vị := (, })" and v2 := (1, 1). Show that Pv,Pv, x = ( vi, v2)( v1, x)v2 for all æ € R². (d) Show that (Pv,Pv, )"x := Pv,Pv, o...o Pv,Pv, x = (v1, v2)²n-1(v1, x)v2 for all æ E (, T. Suppose V; := span{v;}, i = 1,2, and xo := n times R?. (e) Let xn+1 := Pv, Pv, *n, n E No. Compute x1 and x2. (f) Does the sequence {rn}n€No 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 vz such that the sequence {xn}neNo cOnverges.
(a) Define what is meant by a "scalar product" in Rº. span{v1,..., vk}, where {v;}_, are orthonormal vectors, be a linear sub- (b) Let V := space of Rd. What is the projection Py of a vector x E Rd onto V? i=1 (c) Let vị := (, })" and v2 := (1, 1). Show that Pv,Pv, x = ( vi, v2)( v1, x)v2 for all æ € R². (d) Show that (Pv,Pv, )"x := Pv,Pv, o...o Pv,Pv, x = (v1, v2)²n-1(v1, x)v2 for all æ E (, T. Suppose V; := span{v;}, i = 1,2, and xo := n times R?. (e) Let xn+1 := Pv, Pv, *n, n E No. Compute x1 and x2. (f) Does the sequence {rn}n€No 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 vz such that the sequence {xn}neNo cOnverges.
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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handwritten solution accepted for part b
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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