Math 375 - 10/5 - Shawn Zhong

$Best Approximation of Elements • Theorem ○ V: vector space with inner product ○ L⊆V: finite dimensional linear subspace ○ If x∈V then there exists excatly one z∈L ○ that minimizes the distance to x ○ i.e. ∀y∈L, ‖y−x‖≥‖z−x‖ and ○ If y≠z then ‖y−x‖>‖z−x‖ • Solution ○ L is finite dimensional therefore it has a basis ○ Gram-Schmidt says that we can assume the basis is orthonormal ○ i.e. L has a basis {e_1,e_2,…,e_n } where {■8((e_k,e_l )=0&k≠l@(e_k,e_k )=1&∀k)┤ ○ Then z is given by z=(x,e_1 ) e_1+(x,e_2 ) e_2+…+(x,e_n ) e_n ○ Since z is a linear combination of {e_1,…,e_n }, z∈L • Claim ○ x−z is perpendicular to all u∈L ○ i.e. if u∈L then u⊥x−z ○ i.e. (u,x−z)=0 ○ i.e. (u,x)=(u,z) • Proof: (u,x)=(u,z) ○ Let u∈L be given ○ Then {e_1,…e_n } is a basis for L ○ So for certain u_1,…,u_n∈R ○ Calculate (u,x) § (u,x)=(u_1 e_1+…+u_n e_n,x) § =u_1 (e_1,x)+…+u_n (e_n,x) ○ Calculate (u,z) § (u,z)=(u_1 e_1+…+u_n e_n,(x,e_1 ) e_1+…+(x,e_n ) e_n ) § =[u_1 (x_1,e_1 )(e_1,e_1 )+…+u_1 (x_1,e_n )(e_1,e_n )]+… +[u_n (x_1,e_1 )(e_n,e_1 )+…+u_n (x_n,e_n )(e_n,e_n )] § =u_1 (x,e_1 )+u_n (x,e_2 )+…+u_n (x,e_n ) ○ Therefore (u,x)=(u,z) ○ i.e. u⊥x−z, ∀u∈L • Proof: ∀y∈L, ‖y−x‖≥‖z−x‖ ○ Let y∈L be given ○ {█(y−x=(y−z)+(z−x)@y−z⊥z−x)┤ ○ ⇒‖y−x‖^2=‖y−z‖^2+‖z−x‖^2 ○ ⇒‖y−x‖^2≥‖z−x‖^2 ○ ⇒‖y−x‖≥‖z−x‖ ○ Also if y≠z then ‖y−x‖>‖z−x‖ Foorier Series • V={all continuous function f:[0,π]→R • (f,g)\=∫_0^π▒f(x)g(x)dx • Let f_n (x)=sin⁡(nx) • ⇒(f_n,f_m )=∫_0^π▒〖sin⁡(nx) sin⁡(mx)dx〗 x Z…….._T as *U x 11×-21 My ex I TZ _ – _ – — _ — 80 s’ class=’alignnone size-full wp-image-2489 ‘ src=’https://shawnzhong.com/wp-content/uploads/2017/11/img_5a011d7c72599.png’ /></p> </div> <div class="entry-footer"> </div> </div> </article> <div class="post-attributes">  <div class="related-posts"> <h3>Related Posts</h3> <div class="multi-carousel onetone-related-posts owl-carousel owl-theme"> <div class="owl-item"> <div class="post-grid-box"> <div class="img-box figcaption-middle text-center from-left fade-in"> <a href="https://www.shawnzhong.com/2017/10/26/math-375/"> <img src="https://www.shawnzhong.com/wp-content/uploads/2017/10/calculus-applied_378x225.jpg" class="feature-img"/> <div class="img-overlay"> <div class="img-overlay-container"> <div class="img-overlay-content"> <i class="fa fa-link"></i> </div> </div> </div> </a> </div> <div class="img-caption"> <a href="https://www.shawnzhong.com/2017/10/26/math-375/"><h4>Math 375</h4></a> <ul class="entry-meta"> <li class="entry-date"><i class="fa fa-calendar"></i>October 26, 2017</li> <li class="entry-author"><i class="fa fa-user"></i>Shawn</li> </ul> </div> </div> </div> </div> </div>   <div class="comments-area text-left">  <div id="respond" class="respondbg"> <div id="respond" class="comment-respond"> <h3 id="reply-title" class="comment-reply-title">Leave a Reply <small><a rel="nofollow" id="cancel-comment-reply-link" href="/2017/10/26/105/#respond" style="display:none;">Cancel reply</a></small></h3><form action="https://www.shawnzhong.com/wp-comments-post.php" method="post" id="commentform" class="comment-form" novalidate><p class="comment-notes">Your email address will not be published.</p><div class="clear"></div><p class="form-allowed-tags"></p> <section class="comment-form-comment form-group"><div id="comment-textarea"><textarea id="comment" name="comment" placeholder="Message" cols="45" rows="8" class="textarea-comment form-control" aria-required="true"></textarea></div></section><div class="row"><section class="comment-form-author form-group col-md-4"><input id="author" class="input-name form-control" name="author" placeholder="Name" type="text" value="" size="30" aria-required=$

Shawn Zhong

Shawn Zhong

Math 375 – 10/5

Math 375 – 10/5

Search

WeChat Account